Each reference cited herein is expressly incorporated by reference in its entirety. These incorporations are intended to provide written description for aspects of the invention already known, to provide enabling teachings, regardless of field of specialization, and to define useful combinations and contexts of use. Reference citation is not intended to admit prior art status, which is determined by 35 USC 102.
Automated flight control guidance systems are well known. Typically, an aircraft has bilateral symmetry, and displays aerodynamic lift and drag, which are controlled by altering positions of aerodynamic surfaces during flight. A main feature of such craft is one or more pairs of wings on either side of the craft.
The “Frisbee®” is a well-known flying disk toy that is typically formed as an integral rigid tough plastic shell, in a dished shape of about 8-12 inches in diameter. It is thrown with an applied torque, such that it acquires spin as it is launched, i.e., has rotational and translational components to its motion vector. An inverted dish shape provides lift, while drag on the exterior surface(s) may also impact aerodynamics. If launched with a non-neutral roll while thrown, dished disks tend to assume a curved flight path, due to the effect of gravitational force and aerodynamics on the flight path. Other anomalies can also disrupt a straight line flight. An “Aerobie” is a toroid-shaped object with a flat profile, open center, and aerodynamic inner and outer edges to provide stability over a range of speeds, and long distance flight. Electronic projectile rotating disks are described in U.S. Pat. Nos. 9,579,552, 8,444,513, 20130303314, 20110250819, 20060199682, 3,960,379; 4,145,839; 4,301,616; 4,748,366; 4,867,727; 4,869,699; 4,894,038; 4,955,620; 5,032,098; 5,125,862; 5,145,444; 5,256,099; 5,634,839; 5,672,086; 5,893,790; 5,895,308; 5,902,166; 5,975,982; 5,984,753; 6,186,902; 6,206,537; 6,241,362; 6,265,984; 6,270,391; 6,287,193; 6,292,213; 6,304,665; 6,364,614; 6,402,342; 6,404,409; 6,524,073; 6,565,243; 6,604,742; 6,972,659; 7,032,861; 7,082,890; 7,101,293; 7,108,576; 7,187,295; 7,249,732; 7,285,032; 7,541,995; 7,545,994; 7,663,629; 7,775,910; 7,789,520; 7,794,341; 7,850,551; 7,971,823; 8,286,265; 8,322,649; 8,355,410; 8,398,449; 8,417,481; 8,585,476; 8,608,167; 8,678,873; 8,808,100; 8,821,293; 8,849,697; 8,880,378; 8,908,922; 8,909,543; 8,920,287; 8,924,248; 8,924,249; 8,931,144; 8,948,457; 8,982,105; 9,086,782; 9,089,760; 9,143,392; 9,149,695; 9,170,074; 9,237,297; 9,272,782; 9,352,209; 9,352,216; 9,415,263; 9,421,448; 9,464,873; 9,498,689; 9,514,604; 9,560,725; 9,573,035; 9,579,552; 9,582,072; 20020005614; 20020017759; 20020067990; 20020100040; 20020180154; 20030045200; 20040009063; 20040077255; 20040077975; 20040094662; 20040209712; 20040220001; 20040244034; 20050151941; 20050200079; 20050260918; 20060049304; 20060073758; 20060105838; 20060136498; 20060144211; 20060199682; 20060200314; 20060262120; 20060277466; 20060287137; 20070029272; 20070032318; 20070034738; 20070100666; 20070146325; 20070155263; 20070205553; 20070212973; 20070213126; 20070218988; 20070298913; 20080033581; 20080062677; 20080096654; 20080096657; 20080104422; 20080132361; 20080167535; 20080174281; 20080185785; 20080242415; 20080274844; 20090031438; 20090134580; 20090143175; 20090171788; 20090278317; 20090280931; 20090300551; 20100062847; 20100139738; 20100141609; 20100261526; 20100279776; 20100282918; 20100320333; 20110004975; 20110040879; 20110042901; 20110053716; 20110161254; 20120066883; 20120068927; 20120103274; 20120103275; 20120258804; 20130073387; 20130073389; 20130079152; 20130109265; 20130139073; 20130225032; 20130238538; 20130303314; 20130344958; 20140083402; 20140141865; 20140142467; 20140142733; 20140143031; 20140143038; 20140144417; 20140159894; 20140159903; 20140163428; 20140163429; 20140163430; 20140203797; 20140232516; 20140259549; 20140301598; 20140301600; 20140301601; 20140335952; 20140364254; 20140379696; 20150005912; 20150057808; 20150062959; 20150103168; 20150219425; 20150243068; 20150297964; 20150306455; 20150309563; 20150319562; 20150360777; 20160001158; 20160018083; 20160033966; 20160058644; 20160096093; 20160120733; 20160120734; 20160136525; 20160136532; 20160147308; 20160154170; 20160184720; 20160236102; 20160250516; 20160271492; 20160287937; 20160338644; 20160354664; 20160377271; 20170050113; and 20170050116; expressly incorporated herein by reference.
A ball can have 3D radial symmetry, e.g., a basketball, or 2D radial symmetry, e.g., a football. These typically have no intrinsic net lift during flight, though a topspin or backspin can induce lift or negative lift.
It is known to provide “smart bombs” which are passively steered to a target by a steering mechanism. Typically, such smart bombs are not gyroscopically stabilized, since this stabilization tends to undermine steerability. en.wikipedia.org/wiki/Guided_bomb; http://science.howstuffworks.com/smart-bomb.htm;
It is also known to provide smart bullets which are guided or guidable to a target. See U.S. Pat. No. 5,788,178. In many cases, rotation of the bullet projectile is contraindicated. In other cases, a guidance system is provided that permits steering of the rotating bullet projectile. Such guidance systems may have features and structures useful in accordance with the present invention. See, U.S. Pat. Nos. 8,747,241; 8,362,408; 8,319,162; 8,288,698; 8,119,958; 7,999,212; 7,963,442; 7,947,937; 7,891,298; 7,834,301; 7,781,709; 7,692,127; 7,557,433; 7,557,433; 7,412,930; 7,354,017; 7,255,304; 6,608,464; 6,474,593, each of which is expressly incorporated herein by reference in their entirety.
Similarly, spinning satellites also have rotational-angle responsive guidance and control systems, though these typically operate in an environment without aerodynamic influences.
As such, it is known in some applications to provide an angularly referenced flight path adjustment/control system of a rotating object. However, these systems are not generic.
One issue in guiding a rotating object is that the rotation gyroscopically stabilizes the projectile, and attempts to steer the projectile with modification of the axis of rotation, and off the gyroscopically stabilized flight path, result in reactive forces. Any control system for guiding the flight path of a rotating object should therefore compensate for, or predict and respond to, the reactive force resulting from a corrective force (unless the reactive force is consistent with the intended flight path), and in most cases, the corrective force or other response should be synchronized with the rotational angle of the projectile. Each of these presents challenges, especially if the control system is to be self-contained within the projectile of reasonable size, weight and cost. In some cases, where real time feedback is available and the actuator provides a small correction force with respect to the gyroscopic stabilization force, a prediction of reactive forces and effect is not required, though this imposes sometimes significant limitations on the control system. Another issue is that in an inertial guidance system, the static G-forces in a rotating sensor platform may make quantitative analysis of small force perturbations more difficult.
Another issue is fragility. In the case of a Frisbee or ball, the intended use subjects the projectile to harsh conditions, such as dirt, water, saltwater, mud, impacts, manual launch and grabbing, etc. Therefore, even if the projectile meets a first aim of controlled flight path, unless it is also capable of surviving in harsh conditions, it will not meet a second aim of usability and context-appropriateness.
The field of aerodynamics is well studied. In a fixed wing draft, there are a number of control surfaces that are typically employed to steer and control the craft. A rotating disk aircraft lacks obvious dynamic control opportunities, and the gyroscopic effects may be very significant during many types of control attempts.
When a Frisbee is in flight, there are three main external forces acting on it: weight, lift and drag. In addition, the rotation of the Frisbee induces gyroscopic reactive forces in response to perturbations. See, Hanaa Khalid, Moukhtar Khalid, Study of the aerodynamics of translating spinning discs in laminar and turbulent flow, Doctoral dissertation, University of Khartoum (2012); Erynn J. Schroeder, An Aerodynamic Simulation of Disc Flight, Honors Thesis, College of Saint Benedict and Saint John's University (April 2015), expressly incorporated herein by reference in their entirety.
Weight acts along the gravitational vector, generally downward. In a typical Frisbee, the structure is dynamically balanced with the center of mass at the physical center of the disk. Likewise, spherical balls also are typically designed with a center of mass at a physical center of the sphere.
Lift from the airflow along the axis of movement (generally in the y axis direction), opposes the gravitational force, and is what keeps the Frisbee in the air. Lift is caused by differential air pressure and follows what is known as the Bernoulli principle. The curvature of the disc causes the air moving on top of the disc to travel faster and therefore have lower pressure than the air moving on the bottom of the disc. Similar to a plane wing, this difference in pressure above the disc and below the disc results in lift. The center of lift is offset from the center of mass, and thus applies a torque. This, in turn, causes a reaction force when the disk is spinning.
Lift can be calculated using the equation: L=(½)C1AρV2, where A=Area of the cross section; ρ=density of air; V=relative velocity between the air and the Frisbee; and Cl=Coefficient of lift. This depends on the shape of the disc and angle of attack. The angle of attack is the angle between the horizontal and the direction of air flow. Generally, C increases linearly in relation to an increase of an angle of attack, until a certain point where it drops off.
Drag, the result of air resistance, acts parallel to the flow of the air (x direction), opposite the direction of flight, and is what slows the Frisbee down. Drag can also be calculated by using an equation very similar to lift: L=(½)CdAρV2. The one difference in these equations is that the Cd in this equation is the coefficient of drag. Contrary to lift, the coefficient of drag increases quadratically with an increase in the angle of attack. Because the Frisbee typically has an angle of attack, and the drag acts along the axis of movement, the drag is not normal to the axis of rotation.
A relationship between L and D is formed through the lift-to-drag ratio (L/D). The lift-to-drag ratio varies depending on the angle of attack. An angle of attack of around 15° maximizes the lift-to-drag ratio, resulting in the most efficient flight. That is, for a normal Diskcraft disk, a vertical axis of rotation is less efficient than an inclined axis of rotation. Shereef A. Sadek, Saad A. Ragab, Aerodynamics of Rotating Discs, Tenth International Congress of Fluid Dynamics Dec. 16-19, 2010, Ain Soukhna, Red Sea, Egypt, ICFD10-EG-3901.
Potts and Cowther [2], studied the aerodynamics of Frisbee like discs that had an approximately elliptic cross-section and hollowed out underside cavity. The lift and drag coefficients were found to be independent of Reynolds number for the range of tunnel speeds tested. The upper surface flow is characterized by separation at a line (arc) of constant radius on the leading edge rim, followed by reattachment at a line of similar geometry. Trailing vortices detach from the trailing edge rim. The cavity flow is characterized by separation at the leading edge lip, followed by straight-line reattachment.
Badalamenti and Prince [3], studied the effects of endplates on a rotating cylinder in crossflow for a range of Reynolds number. They found that endplates can enhance the lift/drag ratio and increase lift up to a limiting value.
OVERFLOW, is a three-dimensional flow solver that uses structured overset grids. It solves the Navier-Stokes equations in generalized coordinates. It is capable of obtaining time-accurate as well as steady state solutions: ∂Q/∂t+∂E/∂ξ+∂F/∂η+∂G/∂ζ=0, where Q is the vector of conserved variables in generalized coordinates; ξ, η and ζ, are the generalized coordinates; and G, F and G are the total flux (convective and viscous) in the direction of generalized coordinates ξ, η and ζ, respectively.
For a non-rotating disc, the minimum friction velocity occurs at the leading and trailing edges of the disc. At the leading edge, the flow is stagnant and the boundary layer is very thin. At the trailing edge, there is a small separation region where the flow is recirculating at a very low speed. As the rotation speed is increased, λ=0.78, the contour lines become asymmetric and the location of the minimum value moves counterclockwise along the circumference in the negative x-y plane. The maxima of uτ increases with the increase in λ, but remains in the positive x-y plane along the circumference near the leading edge. This is due to the fact that the net relative flow velocity to the disc surface increases in the positive x-y plane while it is reduced in the negative x-y plane. As the rotation speed is increased above the free stream speed, λ=1.51, a double-minima is developed on the upper and lower disc surfaces and the location of the minima moves inward towards the center at −90° with the positive x-axis direction. This is due to the fact that the surface speed matches the flow speed at some radial distance less than the disc radius. Finally, at λ=1.89, a well defined minima is present on the upper and lower surfaces approximately half a radius away from the center. This is a unique feature of this flowfield. On the other hand, the maximum uτ is always located on the disc's circumference since the relative flow velocity is always positive in the positive x-y plane.
For the non-rotating case, the disc behaves like a finite wing with low aspect ratio. Upstream of the disc center on the upper side, the typical leading edge suction, and on the lower side, high pressure region. A common feature of finite wings are the tip vortices, which are present in here as well. Low pressure coefficient can be seen at the disc tips at an approximate angle of 75° from the positive x-direction. These suction peaks are due to the tip vortices formation that is typical of such flowfields. For the rotating-disc, the first noticeable effect due to rotation is the breaking of the flow symmetry. Upstream of the disc center, the pressure coefficient looks similar to the non-rotating case, this might be due to the fact that the boundary layer is very thin and still developing. On the other hand, downstream of the disc center; the rotation effects are more significant. This is evident by the degree of asymmetry in the pressure coefficient contours downstream of the disc center on the upper and lower surfaces. This asymmetry is because the disc rotation affects the surface shear stress distribution along the disc surface as shown above, the surface shear stress being higher in the positive x-y half-plane than in the negative x-y half-plane. This in turn changes the boundary thickness distribution along the disc surface in an asymmetric manner. The asymmetric surface shear stress creates more adverse pressure gradient in the positive x-y half-plane and more favorable pressure gradient in the negative x-y half-plane. Hence, to the incoming inviscid flow the effective disc shape is asymmetric.
As mentioned above, the asymmetric surface shear stress affects the surface pressure distribution, it also affects the shape of the tip vortices and the shed wake. The vortex size is greater for the rotating disc, which also indicates that the separation at the tip occurs earlier than in the non-rotating case. For the rotating disc, the vortex size is reduced and its center is further off the disc surface. For the rotating disc the wake diffuses much faster than the non-rotating case. This suggests that the rotation has the effect of weakening the tip vortices which means that the disc loses lift.
In general, the rotation increases the disc's drag. The side force is directed in the negative y-direction and the moment direction is opposite to the direction of rotation as expected. The increase in drag coefficient is mainly due to increase in the shear stress. For α=0.0°, the viscous drag increases by about 40% and by 50% for α=10.0°. On the other hand, disc rotation increases the pressure drag at α=0.0° by about 380% for the same rotation speed; but it has insignificant influence at α=10.0°. In summary, the drag coefficient increases with disc rotation; however, disc rotation reduces the increase in drag with a. Disc rotation reduces the lift coefficient by about 4% of the non-rotating disc value. In a rotating-disc, the tip vortices are weaker than in the case of nonrotating disc. Disc rotation alters the surface shear stress distribution which in turn changes the boundary layer thickness and structure. The asymmetric surface shear stress creates asymmetric boundary layer thickness, and hence viscous-invicid interaction leads to asymmetric surface pressure distribution. Disc rotation also affects the tip vortices which leads to weakening of the shed vorticity, this in turn reduces the lift coefficient. Drag coefficient increase significantly due to rotation at zero angle of attack but is reduced significantly at 10.0° angle of attack. See, [1] M. M. Zdravkovich, A. J. Flaherty, M. G. Pahle and I. A. Skellhorne, Some aerodynamic aspects of coin-like cylinders, Journal of Fluid Mechanics, 1998, vol 360, pp 73-84; [2] J. R. Potts and W. J. Crowther, The flow over a rotating disc-wing, RAeS Aerodynamics Research Conference, London, UK, 17-18 Apr. 2000; [3] J. R. Potts and W. J. Crowther, Frisbee Aerodynamics, 20th AIAA Applied Aerodynamics Conference, 24-26 Jun. 2002, St. Louis, Mo., expressly incorporated herein by reference in their entirety.
Aerodynamic drag on sports balls may be measured in wind tunnels or during untethered flight. There is a concern that fixtures needed to support the ball in a wind tunnel may influence its drag. Measurements of projectiles under game conditions have been attempted, but are difficult to interpret from the data scatter and are not controlled. In a laboratory study, drag was observed to depend on the ball speed, rotation, roughness, and orientation. A so-called “drag crisis” was observed for a smooth sphere and was comparable to wind tunnel data. Rough sports balls, such as a baseball, showed evidence of a small drag crisis that was less apparent than the smooth sphere. See, Jeffrey R. Kensrud, Lloyd V. Smith, Ball Aerodynamics, 8th Conference of the International Sports Engineering Association (ISEA) In situ drag measurements of sports balls.
Understanding the flight of a ball involves two aerodynamic properties, lift and drag. Lift can be described as the force, not including gravity, on a ball that is directed perpendicular to the ball's trajectory. Drag is the force, Fd, in the direction opposing the ball's flight path [1]. The drag coefficient, Cd, is found from [2] Dd=2Fd/ρAV2, where p is the density of air, A is the cross sectional area of the ball, and V is the speed of the ball. Drag is a function of surface roughness, velocity, and orientation of the ball. It is convenient to use a non-dimensional form of velocity, expressed in terms of the Reynolds number, Re, defined by Rθ=vD/V, where D is diameter of the ball and u is kinematic viscosity of air. The effect of speed on drag can be described by three characteristic ranges. At a low Reynolds number, flow is laminar until separation occurs at roughly 80° from the stagnation point [3]. When the Reynolds number is increased, the separation region becomes turbulent and attaches itself again, carrying the separation point to the backside of the ball to about 120° from the stagnation point. As the separation point moves to the back of the ball, drag is reduced. The reduction in drag can be large and occur over a small change in the Reynolds number. This is referred to as the Drag Crisis, and often occurs at game speeds. As the Reynolds number is increased further, the flow becomes completely turbulent in the boundary layer just after the stagnation point, causing the drag to increase again. The drag crisis has been observed on a smooth sphere for many years. Millikan and Klein [4] explored the drag crisis in their free flight test finding Cd on a smooth sphere as low as 0.08. Achenbach [5] analyzed how surface roughness on a sphere can change behavior in the critical Reynolds region. His data suggested that as surface roughness increased on a sphere, the drag crisis was induced at a lower Reynolds number and was less severe.
Frohlich [6] the effect of surface roughness on the behavior of the drag crisis, and explained that if a baseball acted like a rough sphere, the drag crisis could help explain the behavior of pitched or batted baseballs. Always, Mish, and Hubbard [7] analyzed baseball pitches by triangulating ball location from video taken during the 1996 Summer Olympic Games. Their data showed evidence of a drag crisis with a drag coefficient as low as 0.16. An improved video tracking system involving Major League Baseball (PITCHf/x) [8] showed little evidence of a drag crisis, however. The effect of ball rotation has primarily been directed towards lift. Bearman and Harvey rotated dimpled spheres resembling a golf ball in a wind tunnel [9]. They found drag increased with rotation. Mehta attributed the drag increase with lift effects [10].
The drag crisis is apparent for both the golf ball and smooth sphere. A dimpled surface of the golf ball causes a drag crisis at a lower Reynolds number than the smooth sphere. The optimized dimple pattern of the golf ball helps maintain a relatively large magnitude of its drag crisis. While the stitched balls (baseballs, softballs, and cricket balls) show Cd decreasing with increasing speed, the magnitude of their “drag crisis” is significantly smaller than the smooth sphere and golf ball. Scatter in drag is larger for the balls with raised seams (NCAA baseball and 89 mm softball). The arrangement of the stitches likely plays a role in the drag crisis. The drag crisis is a result of turbulence developing in the boundary layer of the ball. The stitches trip the boundary layer earlier than a smooth sphere, inducing turbulence and moving the flow separation point to the backside of the ball. In the normal orientation, the stitches are close to the flow separation point, which is apparently fixed to the stitch location for balls with raised seams. For smooth spheres, the separation point (and drag) depends on the air speed. For balls with raised seams, the separation point (and drag) depends on variation in ball orientation and air speed. The parallel orientation of the cricket ball exhibited the largest drag crisis; as the stitches are removed from the separation point in this orientation, providing a relatively smooth flow surface. Balls with raised seams (NCAA baseball and 89 mm softball) exhibit a higher relative Cd and a drag crisis at a lower relative Reynolds number. Both trends are consistent for balls of higher relative roughness. Similar to rotating dimpled spheres [9], drag on a baseball increases with spin but with less magnitude. While the effect is small, it is measurable and affects the ball flight. Pitched balls (including knuckleballs) usually have some rotation and therefore do not achieve the low Cd. Thus, in play, one should expect pitched balls to have a drag coefficient between 0.3 and 0.4. However, for pitchers like Tim Wakefield, who can deliver a ball with no rotation, a pronounced drag crisis is more likely to occur, where the Cd could be as low as 0.26. Smooth spheres, golf balls, and balls with flat seams show a strong drag crisis, while raised seam balls show a weak drag crisis. Rotation had an effect on drag, increasing the average Cd on a baseball by 20%. It is difficult to pitch baseballs with no rotation. Hence, most baseball pitches will have a Cd of 0.35 (the average drag observed for rotating balls). See, [1] R. K. Adair, The Physics of Baseball. 3rd ed. (HarperCollins Publishers, New York, 2002), pp. 5-40; [2] C. T. Crowe, F. D. Elger, J. A. Roberson. Engineering Fluid Mechanics. 8th ed. (John Wiley & Sons, Hoboken, 2005) pp 440; [3] R. L. Panton. Incompressible Flow. 3rd ed. (John Wiley, Hoboken, 2005) pp 324-34; [4] C. B. Millikan, A. L. Klein. “The effect of turbulence. An investigation of maximum lift coefficient and turbulence in wind tunnels and in flight,” Aircraft Engineering, Vol 5, 169-74, 1933; [5] E. Achenbach. “The effects of surface roughness and tunnel blockage on flow past spheres,” Journal of Fluid Mechanics, Vol. 65, 113-25 (1974); [6] C. Frohlich. “Aerodynamic drag crisis and its possible effect on the flight of baseballs,” American Journal of Physics, 52, 325-34 (1984); [7] L. W. Alaways, S. P. Mish, M. Hubbard. “Identification of Release Conditions and Aerodynamic Forces in Pitched-Baseball Trajectories,” Journal of Applied Biomechanics, Vol. 17, 63-76 (2001); [8] A. M. Nathan. “Analysis of PITCHf/x Pitched Baseball Trajectories,” 2007. University of Illinois, 8 July, 2009 webusers.npl.illinois.edu/˜a-nathan/pob; [9] P. W. Bearman, and J. K. Harvey. “Golf ball aerodynamics,” Aeronautical Quarterly, Vol. 27, 112-22 (1976); [10] R. Mehta. “Aerodynamics of Sports Balls,” Ann. Rev. Fluid Mechanics, 1985. 17, 151-75 (1985); [11] R. D. Knight. Physics for Scientists and Engineers. Vol. 1 (Pearson Education, Boston, 2007); [12] A. M. Nathan. “The effect of spin on the flight of a baseball.,” American Journal of Physics, 76 (2), 119-24 (2008), expressly incorporated herein by reference in their entirety.
For spinning balls, the Magnus effect is responsible for producing the side or lift force. A negative or reverse Magnus effect can be created. Surface roughness, can be used to generate an asymmetric flow, assisting in the Magnus effect. See, Rabindra D. Mehta and Jani Macari Pallis, “Sports Ball Aerodynamics: Effects of Velocity, Spin and Surface Roughness”, Materials And Science In Sports, Edited by: EH. (Sam) Froes and S. J. Haake, Pgs. 185-197.
Aerodynamics plays a prominent role in defining the flight of a ball that is struck or thrown through the air in almost all ball sports. The main interest is in the fact that the ball can be made to deviate from its initial straight path, resulting in a curved, or sometimes an unpredictable, flight path. Lateral deflection in flight, commonly known as swing, swerve or curve, is well recognized in baseball, cricket, golf, tennis, volleyball and soccer. In most of these sports, the lateral deflection is produced by spinning the ball about an axis perpendicular to the line of flight which generates the Magnus effect. Aerodynamics and hydrodynamics in sports, J. M. Pallis Cislunar Aerospace, Inc., San Francisco, Calif., USA R. D. Mehta Sports Aerodynamics Consultant, Mountain View, Calif., USA. A side force, which makes a ball swing through the air, can also be generated in the absence of the Magnus effect. In one of the cricket deliveries, the ball is released with the seam angled, which creates the boundary layer asymmetry necessary to produce swing. In baseball, volleyball and soccer there is an interesting variation whereby the ball is released without any spin imparted to it. In this case, depending on the seam or stitch orientation, an asymmetric, and sometimes time-varying, flow field can be generated, thus resulting in an unpredictable flight path. Almost all ball games are played in the Reynolds Number range of between about 40,000 to 400,000. The Reynolds number is defined as, Re=Ud/v, where U is the ball velocity, d is the ball diameter and v is the air kinematic viscosity. It is particularly fascinating that, purely through historical accidents, small disturbances on the ball surface, such as the stitching on baseballs and cricket balls, the felt cover on tennis balls and patch-seams on volleyballs and soccer balls, are about the right size to affect boundary layer transition and development in this Re range. See Mehta and Pallis (2001a).
First consider the flight of a smooth sphere through an ideal or inviscid fluid. As the flow accelerates around the front of the sphere, the surface pressure decreases (Bernoulli equation) until a maximum velocity and minimum pressure are achieved half way around the sphere. The reverse occurs over the back part of the sphere so that the velocity decreases and the pressure increases (adverse pressure gradient). In a real viscous fluid such as air, a boundary layer, defined as a thin region of air near the surface, which the sphere carries with it, is formed around the sphere. The boundary layer cannot typically negotiate the adverse pressure gradient over the back part of the sphere and it will tend to peel away or “separate” from the surface. The pressure becomes constant once the boundary layer has separated and the pressure difference between the front and back of the sphere results in a drag force that slows down the sphere. The boundary layer can have two distinct states: “laminar”, with smooth tiers of air passing one on top of the other, or “turbulent”, with the air moving chaotically throughout the layer. The turbulent boundary layer has higher momentum near the wall, compared to the laminar layer, and it is continually replenished by turbulent mixing and transport. It is therefore better able to withstand the adverse pressure gradient over the back part of the sphere and, as a result, separates relatively late compared to a laminar boundary layer. This results in a smaller separated region or “wake” behind the ball and thus less drag. (Note that the turbulent process itself is dissipative, and therefore a mathematical model is complex or inaccurate).
The “transition” from a laminar to a turbulent boundary layer occurs when a critical sphere Reynolds number is achieved. The flow over a sphere can be divided into four distinct regimes (2). The drag coefficient is defined as, CD=D/(0.5 pU2A), where D is the drag force, p is the air density and A is the cross-sectional area of the sphere. In the subcritical regime, laminar boundary layer separation occurs at an angle from the front stagnation point of about 80° and the CD is nearly independent of Re. In the critical regime, the CD drops rapidly and reaches a minimum at the critical Re. The initial drop in CD is due to the laminar boundary layer separation location moving downstream (0 S=95°). At the critical Re, a separation bubble is established at this location whereby the laminar boundary layer separates, transition occurs in the free-shear layer and the layer reattaches to the sphere surface in a turbulent state. The turbulent boundary layer is better able to withstand the adverse pressure gradient over the back part of the ball and separation is delayed to 0 S=120°. In the supercritical regime, transition occurs in the attached boundary layer and the CD increases gradually as the transition and the separation locations creep upstream with increasing Re. A limit is reached in the transcritical regime when the transition location moves all the way upstream, very close to the front stagnation point. The turbulent boundary layer development and separation is then determined solely by the sphere surface roughness, and the CD becomes independent of Re since the transition location cannot be further affected by increasing Re.
Earlier transition of the boundary layer can be induced by “tripping” the laminar boundary layer using a protuberance (e.g. seam on a baseball or cricket ball) or surface roughness (e.g. dimples on a golf ball or fabric cover on a tennis ball). For the smooth sphere, the CD in the subcritical regime is about 0.5 and at the critical Re of about 400,000 the CD drops to a minimum of about 0.07, before increasing again in the supercritical regime. The critical Re, and amount by which the CD drops, both decrease as the surface roughness increases on the sports balls.
In a viscous flow such as air, a sphere that is spinning at a relatively high rate can generate a flow field that is very similar to that of a sphere in an inviscid flow with added circulation. That is because the boundary layer is forced to spin with the ball due to viscous friction, which produces a circulation around the ball, and hence a side force. At more nominal spin rates, such as those encountered on sports balls, the boundary layers cannot negotiate the adverse pressure gradient on the back part of the ball and they tend to separate, somewhere in the vicinity of the sphere apex. The extra momentum applied to the boundary layer on the retreating side of the ball allows it to negotiate a higher pressure rise before separating and so the separation point moves downstream. The reverse occurs on the advancing side and so the separation point moves upstream, thus generating an asymmetric separation and an upward deflected wake.
Following Newton's Third Law of Motion, the upward deflected wake implies a downward (Magnus) force acting on the ball. The dependence of the boundary layer transition and separation locations on Re can either enhance or oppose (even overwhelm) this effect. Since the effective Re on the advancing side of the ball is higher than that on the retreating side, in the subcritical or (especially) supercritical regimes, the separation location on the advancing side will tend to be more upstream compared to that on the retreating side. This is because the CD increases with Re in these regions, thus indicating an upstream moving separation location. However, in the region of the critical Re, a situation can develop whereby the advancing side winds up in the supercritical regime with turbulent boundary layer separation, whereas the retreating side is still in the subcritical regime with laminar separation. This would result in a negative Magnus force, since the turbulent boundary layer on the advancing side will now separate later compared to the laminar layer on the retreating side. Therefore, a sphere with topspin for example, would experience an upward aerodynamic force. So in order to maximize the amount of (positive) Magnus force, it helps to be in the supercritical regime and this can be ensured by lowering the critical Re by adding surface roughness (e.g. dimples on a golf ball).
Two basic aerodynamic principles are used to make a baseball curve in flight: spin about an axis perpendicular to the line of flight and asymmetric boundary-layer separation due to seam location on non-spinning baseballs. Consider a pitch, such as the curveball, where spin is imparted to the baseball in an attempt to alter its flight just enough to fool the batter. The baseball for this particular pitch is released such that it acquires topspin about the horizontal axis. As discussed above, under the right conditions, this results in a (downward) Magnus force that makes the ball curve faster towards the ground than it would under the action of gravity alone. The spin parameter (S) is defined as the ratio of the equatorial velocity at the edge of the ball (V) to its translation velocity (U), hence S=V/U. At such a low Re, the flow over the baseball is expected to be subcritical, but the asymmetric separation and deflected wake flow are clearly evident, thus implying an upward Magnus force. At higher Re, the rotating seam would produce an effective roughness capable of causing transition of the laminar boundary layer. Spin rates of up to 35 revs/sec and speeds of up to 45 m/s (100 mph) are achieved by pitchers in baseball. Alaways (3) analyzed high-speed video data of pitched baseballs (by humans and a machine) and used a parameter estimation technique to determine the lift and drag forces on spinning baseballs. For a nominal pitching velocity range of 17 to 35 m/s (38 to 78 mph) and spin rates of 15 to 70 revs/sec, Alaways (3) gave a CD range of 0.3 to 0.4. This suggests that the flow over a spinning baseball in this parameter range is in the supercritical regime with turbulent boundary layer separation. As discussed above, an asymmetric separation and a positive Magnus force would therefore be obtained in this operating range.
In some wind tunnel measurements of the lateral or lift force (L) on spinning baseballs, Watts and Ferrer (4) concluded that the lift force coefficient, CL [=L/(0.5 pU2A)] was a function of the spin parameter only, for S=0.5 to 1.5, and at most only a weak function of Re, for Re=30,000 to 80,000. Their trends agreed well with Bearman and Harvey's (5) golf ball data obtained at higher Re (up to 240,000) and lower spin parameter range (S=0 to 0.3). Based on these correlations, Watts and Bahill (6) suggested that for spin rates typically encountered in baseball (S<0.4), a straight line relation between CL and S with a slope of unity is a good approximation. Alaways' lift measurements on spinning baseballs obtained for Re=100,000 to 180,000 and S=0.1 to 0.5, were in general agreement with the extrapolated trends of the data due to Watts and Ferrer. However, one main difference was that Alaways found a dependence of seam orientation (2-seam versus 4-seam) on the measured lift coefficient. The CL was higher for the 4-seam case compared to the 2-seam for a given value of S. Watts and Ferrer (4) had also looked for seam orientation effects, but did not find any. Alaways concluded that the seam orientation effects were only significant for S<0.5, and that at higher values of S, the data for the two orientations would collapse, as found by Watts and Ferrer (4). The main difference between these seam orientations is the effective roughness that the flow sees for a given rotation rate. As discussed above, added effective roughness puts the ball deeper into the supercritical regime, thus helping to generate the Magnus force. It is possible that at the higher spin rates (higher values of S), the difference in apparent roughness between the two seam orientations becomes less important.
The main significance of the seam orientation is realized when pitching the fastball. Fastball pitchers wrap their fingers around the ball and release it with backspin so that there is an upward (lift) force on the ball opposing gravity. The fastball would thus drop slower than a ball without spin and since there is a difference between the 2-seam and 4-seam CL, the 4-seam pitch will drop even slower. The maximum measured lift in Alaways' (3) study was equivalent to 48% of the ball's weight.
In golf ball aerodynamics, apart from the lift force (which is generated by the backspin imparted to the ball), the drag and gravitational forces are also important, since the main objective is to “tailor” the flight path of the ball. The lift force is generated due to the Magnus effect and the role of the dimples is to lower the critical Re. The asymmetric separation and downward deflected wake are both apparent and result in an upward lift force on the spinning golf ball. The effect of the dimples is to lower the critical Re. Also, once transition has occurred, the CD for the golf ball does not increase sharply in the supercritical regime, like that for the baseball, for example. This demonstrates that while the dimples are very effective at tripping the laminar boundary layer, they do not cause the thickening of the turbulent boundary layer associated with positive roughness. Bearman and Harvey (5) conducted a comprehensive study of golf ball aerodynamics using a large spinning model mounted in a wind tunnel over a wide range of Re (40,000 to 240,000) and S (0.02 to 0.3). They found that CL increased monotonically with S (from about 0.08 to 0.25), as one would expect, and that the CD also started to increase for S>0.1 (from about 0.27 to 0.32) due to induced drag effects. The trends were found to be independent of Reynolds number for 126,000<Re<238,000. Smits and Smith (8) made wind tunnel measurements on spinning golf balls over the range, 40,000<Re<250,000 and 0.04<S<1.4, covering the range of conditions experienced by the ball when using the full set of clubs. Based on their detailed measurements, which included measurements of the spin decay rate, they proposed a new aerodynamic model of a golf ball in flight. Their measurements were in broad agreement with the observations of Bearman and Harvey, although the new CL measurements were slightly higher (−0.04) and a stronger dependence of CD on the spin parameter was exhibited over the entire S range. A new observation was that for Re>200,000, a second decrease in CD was observed, the first being that due to transition of the boundary layer. Smits and Smith proposed that this could be due to compressibility effects since the local Mach number over the ball reached values of up to 0.5. Note that Bearman and Harvey (5) used a 2.5 times larger model, and so their Mach number was correspondingly lower. Smits and Smith (8) proposed the following model for driver shots in the operating range, 70,000<Re<210,000, 0.08<S<0.2: 192, CD=CD1+CD2S+CD3 sin{π(Re−A1)/A2}, CL=CL1S0.4, Spin Rate Decay=δw/δt [d2/(4U2)]=R1S. Suggested values for the constants are: CD1=0.24, CD2=0.18, CD3=0.06, CL1=0.54, R1=0.00002, A1=90,000 and A2=200,000.
Over the years, several dimple designs and layouts have been tried to improve the golf ball aerodynamics. Bearman and Harvey (5) found that hexagonal dimples, instead of the conventional round ones, improved the performance of the ball since the CL was slightly higher and the CD lower. It was concluded that the hexagonal dimples were perhaps more efficient boundary layer trips by shedding discrete horse-shoe vortices from their straight edges. In seeking to minimize the amount of sideways deflection on an inadvertently sliced drive, a ball was designed (marketed under the name: “Polara”) with regular dimples along a “seam” around the ball and shallower dimples on the sides. The ball is placed on the tee with the seam pointing down the fairway, and if only backspin about the horizontal axis is imparted to it, it will generate roughly the same amount of lift as a conventional ball. However, if the ball is heavily sliced, so that it rotates about a near-vertical axis, the reduced overall roughness increases the critical Re, and hence the sideways (undesirable) deflection is reduced. In an extreme version of this design, where the sides are completely bald, a reverse Magnus effect can occur towards the end of the flight which makes a sliced shot end up to the left of the fairway.
Smits and Smith (3) have reported wind-tunnel measurements of the lift and drag coefficients and the spindown rate on a golf ball (mass=0.04593 kg, radius=0.02134 m). Their data demonstrate that the parameter {acute over (ω)}R2/v2 is an approximately linear function of S, and independent of Reynold's number (for fixed S) in the range (1.0-2.5)×105. Numerically, the spin-down rate is given by golf ball (Smits): dω/dt=−4.0×10−6v2/R2S, (5) with v in mph. For v=100 mph, {acute over (ω)}=ω/23.8, implying a spin-down time constant of 23.8 s. Note that {acute over (ω)} scales with v2S/R2 in the Smits model and with v2RCL/M in the model described above. The two models would therefore appear to be different. However, it should be noted that M scales with R (3) and CL scales with S, so the scaling of {acute over (ω)} with R, v, and S is essentially identical. In effect, the Smits model provides empirical evidence for the more physically based model described above. It is useful to apply the latter model to the golf ball, then use the Smits data to fix the torque parameter k. Putting in the mass and radius appropriate to a golf ball and assuming that CL≈S for a golf ball, an expression identical to Eq. 4 can be derived with the numerical factor equal to 0.0215. This means that for comparable v and k, the time constant for spin decay for a baseball will about 8% larger than that of a golf ball. Using the golf data, we fix the value k=0.020, a factor of 5 smaller than that hypothesized by Adair, corresponding to a factor of 4 larger spin decay time constant.
Tavares et al. have proposed a model for spin decay in which the torque responsible for the spin decay is parametrized as Idω/dt=−RρACMv2, (6) where CM is the so-called coefficient of moment. The spin decay measurements of Tavares, which utilizes a novel radar gun to measure the time-dependent spin, show that CM≈0.012 S. Using I=0.4MR2 and the values of M and R appropriate to a golf ball, Tavares' result can be expressed as golf ball (Tavares): dω/dt=−5.0×10−6v2/R2 S, (7), with v in mph. This equation is identical in form to Eq. 5 with a numerical factor 25% larger. For example, the spin-down time constant for v=100 mph will be 18.9 sec. More generally, if I=αM/R2 and if CM=βS, (6) then one rearrange Eq. 6 to derive an expression for the spin decay time constant τ, τ≡ω/{acute over (ω)}=M/R2 α/πρβv. (8) Therefore for a given v and fixed values of α and β, the spin decay time constant scales with M/R2, allowing a comparison among different spherical balls. For example, a golf ball and baseball have M/R2=101.2 kg/m2 and 109.4 kg/m2, respectively, so that the time constant for a baseball will be about 8% larger than for a golf ball.
See, 1 R. K. Adair, The Physics of Baseball (HarperCollins, New York, 2002) 3rd ed., pp 25-26; 2 G. S. Sawicki, M. Hubbard, and W. Stronge, “How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories,” Am. J. Phys. 71, 1152-1162 (2003); 3 A. J. Smits and D. R. Smith, “A new aerodynamic model of a golf ball in flight,” Science and Golf II, Proceedings of the 1994 World Scientific Congress on Golf, edited by A. J. Cochran and M. R. Farraly (E&FN Spon., London, 1994), pp. 340-347; 4 G. Tavares, K. Shannon, and T. Melvin, “Golf ball spin decay model based on radar measurements,” Science and Golf III, Proceedings of the 1998 World Scientific Congress on Golf, edited by M. R. Farraly and A. J. Cochran (Human Kinetics, Champaign Ill., 1999), pp. 464-472, expressly incorporated by reference in their entirety.
1. R. D. Mehta, “Aerodynamics of Sports Balls,” Annual Review of Fluid Mechanics, 17 (1985), 151-189; 2. E. Achenbach, “Experiments on the Flow Past Spheres at Very High Reynolds Number,” Journal of Fluid Mechanics, 54 (1972), 565-575; 3. L. W. Alaways, “Aerodynamics of the Curve-Ball: An Investigation of the Effects of Angular Velocity on Baseball Trajectories” (Ph.D. dissertation, University of California, Davis, 1998); 4. R. G. Watts and R. Ferrer, “The Lateral Force on a Spinning Sphere: Aerodynamics of a Curveball,” American Journal of Physics, 55 (1987), 40-44; 5. P. W. Bearman and J. K. Harvey, “Golf Ball Aerodynamics,” Aeronautical Quarterly, 27 (1976), 112-122; 6. R. G. Watts and A. T. Bahill, Keep your eye on the ball: Curve balls, Knuckleballs, and Fallacies of Baseball (New York, N.Y.: W. H. Freeman, 2000); 7. R. G. Watts and E. Sawyer, “Aerodynamics of a Knuckleball,” American Journal of Physics, 43 (1975), 960-963; 8. A J. Smits, A. J. D. R. Smith, “A New Aerodynamic Model of a Golf Ball in Flight,” Science and Golf n, ed. A. J. Cochran (London, UK: E. & F. N. Spon, 1994), 341-347; 9. S. J. Haake, S. G. Chadwick, R. J. Dignall, S. Goodwill, and P. Rose, P. “Engineering Tennis Slowing the Game Down,” Sports Engineering, 3 (2) (2000), 131-143; 10. A. J. Cooke, “An Overview of Tennis Ball Aerodynamics,” Sports Engineering, 3 (2) (2000), 123-129; 11. R. D. Mehta and J. M. Pallis, “The Aerodynamics of a Tennis Ball,” submitted to Sports Engineering, January 2001; 12. R. D. Mehta, “Cricket Ball Aerodynamics: Myth Versus Science,” The Engineering of Sport. Research, Development and Innovation, ed. A. J. Subic and S. J. Haake (Oxford, UK: Blackwell Science, 2000), pp. 153-167; 13. T. Asai, T. Akatsuka, and S. Haake “The Physics of Football,” Physics World, 11-6 (1998), 25-27; 14. Q. Wei, R. Lin, and Z. Liu, “Vortex-Induced Dynamic Loads on a Non-Spinning Volleyball,” Fluid Dynamics Research, 3 (1988), 231-237, expressly incorporated herein by reference in their entirety. J. M. Pallis, Aerodynamics and hydrodynamics in sports, Cislunar Aerospace, Inc., San Francisco, Calif., USA Maglischo E. (1982) Swimming Faster: A Comprehensive Guide to the Science of Swimming, Mayfield Publishing Co; Marchaj C (1979) Aero-hydrodynamics Of Sailing. Dodd, Mead & Company, New York; Mehta R. D. (2000) Cricket Ball Aerodynamics: Myth Versus Science. In: The Engineering of Sport. Research, Development and Innovation [eds. A. J. Subic and S. J. Haake], pp. 153-167. Blackwell Science Ltd, Oxford; Mehta R. D. & Pallis J. M. (2001a) Sports Ball Aerodynamics: Effects of Velocity, Spin and Surface roughness. In: Materials and Science in Sports. [eds. F. H. Froes and S. J. Haake], pp. 185-197. The Minerals, Metals and Materials Society [TMS], Warrendale, USA; Mehta R. D. & Pallis J. M. (2001 b) The Aerodynamics of a Tennis Ball. Sports Engineering, 4, 177-189; Pallis J., Banks D. & Okamoto K. (2000) 3D Computational Fluid Dynamics in Competitive Sail, Yacht and Windsurf Design. In: The Engineering of Sport. Research, Development and Innovation [eds. A. J. Subic and S. J. Haake], pp. 153167. Blackwell Science Ltd, Oxford; Smits A. J. & Smith D. R. (1994) A New Aerodynamic Model of a Golf Ball in Flight. In: Science and Golf II (ed. A. J. Cochran), pp. 341-347. E. & F. N. Spon, London; Watts R. G. & Sawyer E. (1975) Aerodynamics of a Knuckleball. American Journal of Physics, 43, 960-963; Watts R. G. & Ferrer R. (1987) The Lateral Force on a Spinning Sphere: Aerodynamics of a Curveball. American Journal of Physics, 55, 40-44; Whidden T. & Levitt M. (1990) The Art and Science of Sails. St. Martin's Press, New York, each of which is expressly incorporated herein by reference in their entirety.
Aerodynamics of Winged Craft
Aerodynamic force are well studied. (see en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft), expressly incorporated herein by reference. Refer to FIG. 1 for axis definitions.
Components of the Aerodynamic Force
The expression to calculate the aerodynamic force is:FA=∫Σ(−Δpn+f)dσ,                 where:        Δp Difference between static pressure and free current pressure        n≡outer normal vector of the element of area        f≡tangential stress vector of the air on the body        Σ≡adequate reference surface projected on wind axes, obtaining:FA=−(iwD+jwQ+kwL)        
where: D≡Drag; Q≡Lateral force; and L≡Lift
Aerodynamic Coefficients
            Dynamic      ⁢                          ⁢      pressure      ⁢                          ⁢      of      ⁢                          ⁢      the      ⁢                          ⁢      free      ⁢                          ⁢      current        ≡    q    =            1      2        ⁢    ρ    ⁢                  ⁢          V      2      
Proper reference surface (wing surface, in case of planes)≡S
                    ⁢                            Pressure          ⁢                                          ⁢          coefficient                ≡                  C          p                    =                        p          -                      p            ∞                          q                                ⁢                            Friction          ⁢                                          ⁢          coefficient                ≡                  C          f                    =              f        q                                Drag        ⁢                                  ⁢        coefficient            ≡              C        d              =                  D                  q          ⁢                                          ⁢          S                    =                        -                      1            S                          ⁢                              ∫            Σ                    ⁢                                    [                                                                    (                                          -                                              C                        p                                                              )                                    ⁢                                      n                    ·                                          i                      w                                                                      +                                                      C                    f                                    ⁢                                      t                    ·                                          i                      w                                                                                  ]                        ⁢            d            ⁢                                                  ⁢            σ                                                  Lateral        ⁢                                  ⁢        force        ⁢                                  ⁢        coefficient            ≡              C        Q              =                  Q                  q          ⁢                                          ⁢          S                    =                        -                      1            S                          ⁢                              ∫            Σ                    ⁢                                    [                                                                    (                                          -                                              C                        p                                                              )                                    ⁢                                      n                    ·                                          j                      w                                                                      +                                                      C                    f                                    ⁢                                      t                    ·                                          j                      w                                                                                  ]                        ⁢            d            ⁢                                                  ⁢            σ                                                  Lift        ⁢                                  ⁢        coefficient            ≡              C        LO              =                  L                  q          ⁢                                          ⁢          S                    =                        -                      1            S                          ⁢                              ∫            Σ                    ⁢                                    [                                                                    (                                          -                                              C                        p                                                              )                                    ⁢                                      n                    ·                                          k                      w                                                                      +                                                      C                    f                                    ⁢                                      t                    ·                                          k                      w                                                                                  ]                        ⁢            d            ⁢                                                  ⁢            σ                              
It is necessary to know Cp and Cf in every point on the considered surface.
Dimensionless Parameters and Aerodynamic Regimes
In absence of thermal effects, there are three remarkable dimensionless numbers:
                    Compressibility        ⁢                                  ⁢        of        ⁢                                  ⁢        the        ⁢                                  ⁢        flow        ⁢                  :                ⁢                                  ⁢        Mach        ⁢                                  ⁢        number            ≡      M        =          V      a                          Viscosity        ⁢                                  ⁢        of        ⁢                                  ⁢        the        ⁢                                  ⁢        flow        ⁢                  :                ⁢                                  ⁢        Reynolds        ⁢                                  ⁢        number            ≡              R        ⁢                                  ⁢        e              =                  ρ        ⁢                                  ⁢        V        ⁢                                  ⁢        l            μ                          Rarefaction        ⁢                                  ⁢        of        ⁢                                  ⁢        the        ⁢                                  ⁢        flow        ⁢                  :                ⁢                                  ⁢        Knudsen        ⁢                                  ⁢        number            ≡              K        ⁢                                  ⁢        n              =          λ      l      where: α=√{square root over (kRθ)} speed of sound; R≡gas constant by mass unity; θ≡absolute temperature, and
  λ  =                    μ        ρ            ⁢                        π                      2            ⁢                                                  ⁢            R            ⁢                                                  ⁢            θ                                =                            M                      R            ⁢                                                  ⁢            e                          ⁢                                            k              ⁢                                                          ⁢              π                        2                              ≡              mean        ⁢                                  ⁢        free        ⁢                                  ⁢                  path          .                    
According to λ there are three possible rarefaction grades and their corresponding motions are called:
            Continuum      ⁢                          ⁢      current      ⁢                          ⁢              (                  negligible          ⁢                                          ⁢          rarefaction                )            ⁢              :            ⁢                          ⁢              M                  R          ⁢                                          ⁢          e                      ⪡    1              Transition      ⁢                          ⁢              current        ⁢                                                  ⁢                                                (                  moderate          ⁢                                          ⁢          rarefaction                )            ⁢              :            ⁢                          ⁢              M                  R          ⁢                                          ⁢          e                      ≈    1              Free      ⁢                          ⁢      molecular      ⁢                          ⁢      current      ⁢                          ⁢              (                  high          ⁢                                          ⁢          rarefaction                )            ⁢              :            ⁢                          ⁢              M                  R          ⁢                                          ⁢          e                      ⪢    1  
The motion of a body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body viscosity will be negligible. However viscosity effects will have to be considered when analyzing the flow in the nearness of the boundary layer.
Depending on the compressibility of the flow, different kinds of currents can be considered: Incompressible subsonic current: 0<M<0.3; Compressible subsonic current: 0.3<M 0.8; Transonic current: 0.8<Mm<1.2; Supersonic current: 1.2<M<5; and Hypersonic current: 5<M
In most cases, the human-powered embodiments according to the present technology fall into the incompressible subsonic current regime, though in a baseball embodiment, the bottom end of the compressible subsonic range may be approached on some surfaces due to spin.
In an embodiment with mechanism-powered rotation, higher rotational speeds can be achieved, and thus parts of the system can reach the compressible subsonic range or above.
Drag Coefficient Equation and Aerodynamic Efficiency
If the geometry of the body is fixed and in case of symmetric flight (3=0 and Q=0), pressure and friction coefficients are functions depending on: Cp=Cp(α, M, Re, P); Cf=Cf(α, M, Re, P) where: α≡angle of attack, and P≡considered point of the surface.
Under these conditions, drag and lift coefficient are functions depending exclusively on the angle of attack of the body and Mach and Reynolds numbers. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well.
  {                                                        C              D                        =                                          C                D                            (                              α                ,                M                ,                Re                            )                                                                                      C              L                        =                                          C                L                            (                              α                ,                M                ,                Re                            )                                                                        E            =                                          E                (                                  α                  ,                  M                  ,                  Re                                )                            =                                                C                  L                                                  C                  D                                                                          ⁢                     
It is also possible to get the dependency of the drag coefficient respect to the lift coefficient. This relation is known as the drag coefficient equation:CD=CD(CL,M,Re)≡drag coefficient equation.
The aerodynamic efficiency has a maximum value, Emax, respect to CL where the tangent line from the coordinate origin touches the drag coefficient equation plot.
The drag coefficient, CD, can be decomposed in two ways. First typical decomposition separates pressure and friction effects:
      C    D    =            C      Df        +                  C        Dp            ⁢              {                                                                              C                  Df                                =                                                      D                                          s                      ⁢                                                                                          ⁢                      S                                                        =                                                            -                                              1                        S                                                              ⁢                                                                  ∫                        Σ                                            ⁢                                                                        C                          f                                                ⁢                                                  t                          ·                                                      i                            w                                                                          ⁢                                                                                                  ⁢                        d                        ⁢                                                                                                  ⁢                        σ                                                                                                                                                                                      C                  Dp                                =                                                      D                                          s                      ⁢                                                                                          ⁢                      S                                                        =                                                            -                                              1                        S                                                              ⁢                                                                  ∫                        Σ                                            ⁢                                                                        (                                                      -                                                          C                              p                                                                                )                                                ⁢                                                  n                          ·                                                      i                            w                                                                          ⁢                                                                                                  ⁢                        d                        ⁢                                                                                                  ⁢                        σ                                                                                                                                    
There's a second typical decomposition taking into account the definition of the drag coefficient equation. This decomposition separates the effect of the lift coefficient in the equation, obtaining two terms CD0 and CD1. CD0 is known as the parasitic drag coefficient and it is the base drag coefficient at zero lift. CDi is known as the induced drag coefficient and it is produced by the body lift:
      C    D    =            C              D        ⁢                                  ⁢        0              +                  C        Di            ⁢              {                                                                              C                                      D                    ⁢                                                                                  ⁢                    0                                                  =                                                      (                                          C                      D                                        )                                                        C                    L                                                                                                                          C                Di                                                        
Parabolic and Generic Drag Coefficient
A good attempt for the induced drag coefficient is to assume a parabolic dependency of the liftCDi=kCL2⇒CD=CD0+kCL2 
Aerodynamic efficiency is now calculated as:
  E  =                    C        L                              C                      D            ⁢                                                  ⁢            0                          +                  k          ⁢                                          ⁢                      C            L            2                                ⇒          {                                                                  E                max                            =                              1                                  2                  ⁢                                                            k                      ⁢                                                                                          ⁢                                              C                                                  D                          ⁢                                                                                                          ⁢                          0                                                                                                                                                                                                                              (                                      C                    L                                    )                                                  E                  max                                            =                                                                    C                                          D                      ⁢                                                                                          ⁢                      0                                                        k                                                                                                                                          (                                      C                    Di                                    )                                                  E                  max                                            =                              C                                  D                  ⁢                                                                          ⁢                  0                                                                        
If the configuration of a plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane:CDmin=(CD)CL=0=CD0 
In case the configuration is asymmetrical respect to the XY plane, however, minimum drag differs from the parasitic drag. On these cases, a new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value.CDmin=CDM≠(CD)CL=0; CD=CDM+k(CL−CLM)2 
Variation of Parameters with the Mach Number
The Coefficient of pressure varies with Mach number by the relation:
            C      p        =                  C                  p          ⁢                                          ⁢          0                                                            1            -                          M              ∞              2                                                      ,where
Cp is the compressible pressure coefficient
Cp0 is the incompressible pressure coefficient
M∞ is the freestream Mach number.
This relation is reasonably accurate for 0.3<M<0.7 and when M=1 it becomes ∞ which is impossible physical situation and is called Prandtl-Glauert singularity.
Dynamic Stability and Control
Longitudinal Modes
It is common practice to derive a fourth order characteristic equation to describe the longitudinal motion, and then factorize it approximately into a high frequency mode and a low frequency mode. The approach adopted here is using qualitative knowledge of aircraft behavior to simplify the equations from the outset, reaching the result by a more accessible route.
The two longitudinal motions (modes) are called the short period pitch oscillation (SPPO), and the phugoid.
Short-Period Pitch Oscillation
A short input (in control systems terminology an impulse) in pitch (generally via the elevator in a standard configuration fixed-wing aircraft) will generally lead to overshoots about the trimmed condition. The transition is characterized by a damped simple harmonic motion about the new trim. There is very little change in the trajectory over the time it takes for the oscillation to damp out.
Generally, this oscillation is high frequency (hence short period) and is damped over a period of a few seconds. A real-world example would involve a pilot selecting a new climb attitude, for example 5° nose up from the original attitude. A short, sharp pull back on the control column may be used, and will generally lead to oscillations about the new trim condition. If the oscillations are poorly damped the aircraft will take a long period of time to settle at the new condition, potentially leading to Pilot-induced oscillation. If the short period mode is unstable it will generally be impossible for the pilot to safely control the aircraft for any period of time.
This damped harmonic motion is called the short period pitch oscillation, it arises from the tendency of a stable aircraft to point in the general direction of flight. It is very similar in nature to the weathercock mode of missile or rocket configurations. The motion involves mainly the pitch attitude θ (theta) and incidence α (alpha). The direction of the velocity vector, relative to inertial axes is θ−α. The velocity vector is: uf=U cos(θ−α); wf=U sin(θ−α), where uf, wf are the inertial axes components of velocity. According to Newton's Second Law, the accelerations are proportional to the forces, so the forces in inertial axes are:
                              X          f                =                              m            ⁢                                          d                ⁢                                                                  ⁢                                  u                  f                                                            d                ⁢                                                                  ⁢                t                                              =                                    m              ⁢                                                d                  ⁢                                                                          ⁢                  U                                                  d                  ⁢                                                                          ⁢                  t                                            ⁢                              cos                ⁡                                  (                                      θ                    -                    α                                    )                                                      -                          m              ⁢                                                          ⁢              U              ⁢                                                d                  ⁡                                      (                                          θ                      -                      α                                        )                                                                    d                  ⁢                                                                          ⁢                  t                                            ⁢                              sin                ⁡                                  (                                      θ                    -                    α                                    )                                                                                                      Z          f                =                              m            ⁢                                          d                ⁢                                                                  ⁢                                  w                  f                                                            d                ⁢                                                                  ⁢                t                                              =                                    m              ⁢                                                d                  ⁢                                                                          ⁢                  U                                                  d                  ⁢                                                                          ⁢                  t                                            ⁢                              sin                ⁡                                  (                                      θ                    -                    α                                    )                                                      -                          m              ⁢                                                          ⁢              U              ⁢                                                d                  ⁡                                      (                                          θ                      -                      α                                        )                                                                    d                  ⁢                                                                          ⁢                  t                                            ⁢                              cos                ⁡                                  (                                      θ                    -                    α                                    )                                                                        
where m is the mass. By the nature of the motion, the speed variation
  m  ⁢            d      ⁢                          ⁢      U              d      ⁢                          ⁢      t      is negligible over the period of the oscillation, so:
            X      f        =                  -        m            ⁢                          ⁢      U      ⁢                        d          ⁡                      (                          θ              -              α                        )                                    d          ⁢                                          ⁢          t                    ⁢              sin        ⁡                  (                      θ            -            α                    )                      ;            Z      f        =                  -        m            ⁢                          ⁢      U      ⁢                        d          ⁡                      (                          θ              -              α                        )                                    d          ⁢                                          ⁢          t                    ⁢              cos        ⁡                  (                      θ            -            α                    )                    
But the forces are generated by the pressure distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axis forces into wind axes. Also, for fixed wing craft, we are generally only concerned with the force along the z-axis:Z=−Zf cos(θ−α)+Xf sin(θ−α),or
  Z  =            -      m        ⁢                  ⁢    U    ⁢                            d          ⁡                      (                          θ              -              α                        )                                    d          ⁢                                          ⁢          t                    .      
In words, the wind axis force is equal to the centripetal acceleration.
The moment equation is the time derivative of the angular momentum:
      M    =          B      ⁢                                                        ⁢                                    d              2                        ⁢            θ                                    d          ⁢                                          ⁢                      t            2                                ,
where M is the pitching moment, and B is the moment of inertia about the pitch axis. Let:
                    d        ⁢                                  ⁢        θ                    d        ⁢                                  ⁢        t              =    q    ,the pitch rate. The equations of motion, with all forces and moments referred to wind axes are, therefore:
                    d        ⁢                                  ⁢        α                    d        ⁢                                  ⁢        t              =          q      +              Z                  m          ⁢                                          ⁢          U                      ,                    d        ⁢                                  ⁢        q                    d        ⁢                                  ⁢        t              =                  M        B            .      
In the typical fixed wing craft example, we are only concerned with perturbations in forces and moments, due to perturbations in the states a and q, and their time derivatives. These are characterized by stability derivatives determined from the flight condition. The possible stability derivatives are:
Zα Lift due to incidence, this is negative because the z-axis is downwards whilst positive incidence causes an upwards force.
Zq Lift due to pitch rate, arises from the increase in tail incidence, hence is also negative, but small compared with Zα.
Mα Pitching moment due to incidence—the static stability term. Static stability requires this to be negative.
Mq Pitching moment due to pitch rate—the pitch damping term, this is always negative.
Since the trailing portion of the craft operates in the leading portion flowfield, changes in the leading portion incidence cause changes in the downwash, but there is a delay for the change in flowfield to affect the trailing portion lift, this is represented as a moment proportional to the rate of change of incidence: Mα
Increasing the leading portion incidence without increasing the trailing portion incidence produces a nose up moment, so Mα is expected to be positive.
The equations of motion, with small perturbation forces and moments become:
                    d        ⁢                                  ⁢        α                    d        ⁢                                  ⁢        t              =                            (                      1            +                                          Z                q                                            m                ⁢                                                                  ⁢                U                                              )                ⁢        q            +                                    Z            α                                m            ⁢                                                  ⁢            U                          ⁢        α              ;                    d        ⁢                                  ⁢        q                    d        ⁢                                  ⁢        t              =                                        M            q                    B                ⁢        q            +                                    M            α                    B                ⁢        α            +                                    M                          α              .                                B                ⁢                  α          .                    
These may be manipulated to yield as second order linear differential equation in α:
                              d          2                ⁢        α                    d        ⁢                                  ⁢                  t          2                      -                  (                                            Z              α                                      m              ⁢                                                          ⁢              U                                +                                    M              q                        B                    +                                    (                              1                +                                                      Z                    q                                                        m                    ⁢                                                                                  ⁢                    U                                                              )                        ⁢                                          M                                  α                  .                                            B                                      )            ⁢                        d          ⁢                                          ⁢          α                          d          ⁢                                          ⁢          t                      +                  (                                                            Z                α                                                                                              ⁢                                  m                  ⁢                                                                          ⁢                  U                                                      ⁢                                          M                q                            B                                -                                                    M                α                            B                        ⁢                          (                              1                +                                                      Z                    q                                                        m                    ⁢                                                                                  ⁢                    U                                                              )                                      )            ⁢      α        =  0
This represents a damped simple harmonic motion.
We should expect
      Z    q        m    ⁢                  ⁢    U  to be small compared with unity, so the coefficient of α (the ‘stiffness’ term) will be positive, provided
      M    α    <                    Z        α                    m        ⁢                                  ⁢        U              ⁢                  M        q            .      This expression is dominated by Mα, which defines the longitudinal static stability of the aircraft, it must be negative for stability. The damping term is reduced by the downwash effect, and it is difficult to design an aircraft with both rapid natural response and heavy damping. Usually, the response is underdamped but stable.
Phugoid
If the control is held fixed, the aircraft will not maintain straight and level flight, but will start to dive, level out and climb again. It will repeat this cycle until the pilot intervenes. This long period oscillation in speed and height is called the phugoid mode. This is analyzed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the climb angle γ (gamma) and speed. The small perturbation equations of motion are:
            m      ⁢                          ⁢      U      ⁢                        d          ⁢                                          ⁢          γ                          d          ⁢                                          ⁢          t                      =          -      Z        ,which means the centripetal force is equal to the perturbation in lift force.
For the speed, resolving along the trajectory:
          ⁢                    m        ⁢                              d            ⁢                                                  ⁢            u                                d            ⁢                                                  ⁢            t                              =              X        =                  m          ⁢                                          ⁢          g          ⁢                                          ⁢          γ                      ,  where g is the acceleration due to gravity at the earth's surface. The acceleration along the trajectory is equal to the net x-wise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the climb angle, so only Xu and Zu need be considered. Xu is the drag increment with increased speed, it is negative, likewise Zu is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the z-axis.
The equations of motion become:
            m      ⁢                          ⁢      U      ⁢                        d          ⁢                                          ⁢          γ                          d          ⁢                                          ⁢          t                      =                  -                  Z          u                    ⁢      u        ;          ⁢            m      ⁢                        d          ⁢                                          ⁢          u                          d          ⁢                                          ⁢          t                      =                            X          u                ⁢        u            =              m        ⁢                                  ⁢        g        ⁢                                  ⁢                  γ          .                    
These may be expressed as a second order equation in climb angle or speed perturbation:
                                          d            ⁢                                                          2                ⁢        u                    d        ⁢                                  ⁢                  t          2                      -                            X          u                m            ⁢                        d          ⁢                                          ⁢          u                          d          ⁢                                          ⁢          t                      -                                        Z            u                    ⁢          g                          m          ⁢                                          ⁢          U                    ⁢      u        =  0
Now lift is very nearly equal to weight:
      Z    =                            1          2                ⁢        ρ        ⁢                                  ⁢                  U                      C            L                    2                ⁢                  S          w                    =      W        ,where ρ is the air density, Sw is the wing area, W the weight and CL is the lift coefficient (assumed constant because the incidence is constant), we have, approximately:
      Z    u    =                    2        ⁢                                  ⁢        W            U        =                            2          ⁢                                          ⁢          m          ⁢                                          ⁢          g                U            .      
The period of the phugoid, T, is obtained from the coefficient of u:
                    2        ⁢                                  ⁢        π            T        =                            2          ⁢                                          ⁢                      g            2                                    U          2                      ,                    or:            ⁢                          ⁢      T        =                            2          ⁢                                          ⁢          π          ⁢                                          ⁢          U                                      2                    ⁢          g                    .      Since the lift is very much greater than the drag, the phugoid is at best lightly damped. Heavy damping of the pitch rotation or a large rotational inertia increase the coupling between short period and phugoid modes, so that these will modify the phugoid.
Equations of Motion
Dutch Roll is induced in a fixed wing aircraft by an impulse input to the rudder. The yaw plane translational equation, as in the pitch plane, equates the centripetal acceleration to the side force:
                    d        ⁢                                  ⁢        β                    d        ⁢                                  ⁢        t              =                  Y                  m          ⁢                                          ⁢          U                    -      r        ,where β (beta) is as before sideslip angle, Y the side force and r the yaw rate.
The moment equations are a bit trickier. The trim condition is with the aircraft at an angle of attack with respect to the airflow, the body x-axis does not align with the velocity vector, which is the reference direction for wind axes. In other words, wind axes are not principal axes (the mass is not distributed symmetrically about the yaw and roll axes). Consider the motion of an element of mass in position −z,x in the direction of the y-axis, i.e. into the plane of the paper.
If the roll rate is p, the velocity of the particle is:v=−pz+xr. 
Made up of two terms, the force on this particle is first the proportional to rate of v change, the second is due to the change in direction of this component of velocity as the body moves. The latter terms gives rise to cross products of small quantities (pq,pr,qr), which are later discarded. In this analysis, they are discarded from the outset for the sake of clarity. In effect, we assume that the direction of the velocity of the particle due to the simultaneous roll and yaw rates does not change significantly throughout the motion.
With this simplifying assumption, the acceleration of the particle becomes:
            d      ⁢                          ⁢      v              d      ⁢                          ⁢      t        =                    -                              d            ⁢                                                  ⁢            p                                d            ⁢                                                  ⁢            t                              ⁢      z        +                            d          ⁢                                          ⁢          r                          d          ⁢                                          ⁢          t                    ⁢      x      
The yawing moment is given by:
      δ    ⁢                  ⁢    m    ⁢                  ⁢    x    ⁢                  d        ⁢                                  ⁢        v                    d        ⁢                                  ⁢        t              =                    -                              d            ⁢                                                  ⁢            p                                d            ⁢                                                  ⁢            t                              ⁢      x      ⁢                          ⁢      z      ⁢                          ⁢      δ      ⁢                          ⁢      m        +                            d          ⁢                                          ⁢          r                          d          ⁢                                          ⁢          t                    ⁢              x        2            ⁢      δ      ⁢                          ⁢      m      
There is an additional yawing moment due to the offset of the particle in the y direction:
            d      ⁢                          ⁢      r              d      ⁢                          ⁢      t        ⁢      y    2    ⁢  δ  ⁢          ⁢      m    .  
The yawing moment is found by summing over all particles of the body:
  N  =                              -                                    d              ⁢                                                          ⁢              p                                      d              ⁢                                                          ⁢              t                                      ⁢                  ∫                      x            ⁢                                                  ⁢            z            ⁢                                                  ⁢            d            ⁢                                                  ⁢            m                              +                                    d            ⁢                                                  ⁢            r                                d            ⁢                                                  ⁢            t                          ⁢                  ∫                      x            2                              +                        y          2                ⁢        d        ⁢                                  ⁢        m              =                            -          E                ⁢                              d            ⁢                                                  ⁢            p                                d            ⁢                                                  ⁢            t                              +              C        ⁢                              d            ⁢                                                  ⁢            r                                d            ⁢                                                  ⁢            t                              
where N is the yawing moment, E is a product of inertia, and C is the moment of inertia about the yaw axis. A similar reasoning yields the roll equation:
      L    =                  A        ⁢                              d            ⁢                                                  ⁢            p                                d            ⁢                                                  ⁢            t                              -              E        ⁢                              d            ⁢                                                  ⁢            r                                d            ⁢                                                  ⁢            t                                ,where L is the rolling moment and A the roll moment of inertia.
As noted above, in a rotating disk craft, the base yaw is not zero, but remains relatively constant. Due to gyroscopic effect, any change in yaw axis with respect to an inertial reference will create forces along the other axes, thus modifying the behavior with respect to fixed wing aircraft.
Lateral and Longitudinal Stability Derivatives
The states are β (sideslip), r (yaw rate), and p (roll rate), with moments N (yaw) and L (roll), and force Y (sideways). There are nine stability derivatives relevant to this motion, the following explains how they originate.
Yβ Side force due to side slip (in absence of yaw). Sideslip generates a sideforce (from the fin and the fuselage of a fixed wing aircraft). In addition, if the wing has dihedral, side slip at a positive roll angle increases incidence on the starboard wing and reduces it on the port side, resulting in a net force component directly opposite to the sideslip direction. Sweep back of the wings has the same effect on incidence, but since the wings are not inclined in the vertical plane, backsweep alone does not affect Yβ However, anhedral may be used with high backsweep angles in high performance aircraft to offset the wing incidence effects of sideslip. This does not reverse the sign of the wing configuration's contribution to Yβ (compared to the dihedral case).
Yp Side force due to roll rate. Roll rate causes incidence at the fin, which generates a corresponding side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing has dihedral, this will result in a side force momentarily opposing the resultant sideslip tendency. Anhedral wing and or stabilizer configurations can cause the sign of the side force to invert if the fin effect is swamped.
Yr Side force due to yaw rate. Yawing generates side forces due to incidence at the rudder, fin and fuselage.
Nβ Yawing moment due to sideslip forces. Sideslip in the absence of rudder input causes incidence on the fuselage and empennage, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions. Under sideslip conditions at a given roll angle Nβ will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight.
Np Yawing moment due to roll rate. Roll rate generates fin lift causing a yawing moment and also differentially alters the lift on the wings, thus affecting the induced drag contribution of each wing, causing a (small) yawing moment contribution. Positive roll generally causes positive Np values unless the empennage is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on Np because the wing axis is normally closely aligned with the center of gravity.
Nr Yawing moment due to yaw rate. Yaw rate input at any roll angle generates rudder, fin and fuselage force vectors which dominate the resultant yawing moment. Yawing also increases the speed of the outboard wing whilst slowing down the inboard wing, with corresponding changes in drag causing a (small) opposing yaw moment. Nr opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input.
Lβ Rolling moment due to sideslip. A positive sideslip angle generates empennage incidence which can cause positive or negative roll moment depending on its configuration. For any non-zero sideslip angle dihedral wings causes a rolling moment which tends to return the aircraft to the horizontal, as does back swept wings. With highly swept wings the resultant rolling moment may be excessive for all stability requirements and anhedral could be used to offset the effect of wing sweep induced rolling moment.
Lr Rolling moment due to yaw rate. Yaw increases the speed of the outboard wing whilst reducing speed of the inboard one, causing a rolling moment to the inboard side. The contribution of the fin normally supports this inward rolling effect unless offset by anhedral stabilizer above the roll axis (or dihedral below the roll axis).
Lp Rolling moment due to roll rate. Roll creates counter rotational forces on both starboard and port wings whilst also generating such forces at the empennage. These opposing rolling moment effects have to be overcome by the aileron input in order to sustain the roll rate. If the roll is stopped at a non-zero roll angle the upward L, rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the downward Lr rolling moment resulting from sideslip induced yaw rate. Longitudinal stability could be ensured or improved by minimizing the latter effect.
Lateral Modes
With a symmetrical rocket or missile, the directional stability in yaw is the same as the pitch stability; it resembles the short period pitch oscillation, with yaw plane equivalents to the pitch plane stability derivatives. For this reason, pitch and yaw directional stability are collectively known as the ‘weathercock’ stability of the missile.
Fixed wing aircraft lack the symmetry between pitch and yaw, so that directional stability in yaw is derived from a different set of stability derivatives, the yaw plane equivalent to the short period pitch oscillation, which describes yaw plane directional stability is called Dutch roll. Unlike pitch plane motions, the lateral modes involve both roll and yaw motion.
Dutch Roll. Since Dutch roll is a handling mode, analogous to the short period pitch oscillation, any effect it might have on the trajectory may be ignored. The body rate r is made up of the rate of change of sideslip angle and the rate of turn. Taking the latter as zero, assuming no effect on the trajectory, for the limited purpose of studying the Dutch roll:
            d      ⁢                          ⁢      β              d      ⁢                          ⁢      t        =      -          r      .      
The yaw and roll equations, with the stability derivatives become:
                              C          ⁢                                    d              ⁢                                                          ⁢              r                                      d              ⁢                                                          ⁢              t                                      -                  E          ⁢                                    d              ⁢                                                          ⁢              p                                      d              ⁢                                                          ⁢              t                                          =                                    N            β                    ⁢          β                -                              N            r                    ⁢                                    d              ⁢                                                          ⁢              β                                      d              ⁢                                                          ⁢              t                                      +                              N            p                    ⁢          p          ⁢                                          ⁢                      (            yaw            )                                ;                      A        ⁢                              d            ⁢                                                  ⁢            p                                d            ⁢                                                  ⁢            t                              -              E        ⁢                              d            ⁢                                                  ⁢            r                                d            ⁢                                                  ⁢            t                                =                            L          β                ⁢        β            -                        L          r                ⁢                              d            ⁢                                                  ⁢            β                                d            ⁢                                                  ⁢            t                              +                        L          p                ⁢        p        ⁢                                  ⁢                  (          roll          )                    
The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become:
                    -        C            ⁢                                    d            2                    ⁢          β                          d          ⁢                                          ⁢                      t            2                                =                            N          β                ⁢        β            -                        N          r                ⁢                              d            ⁢                                                  ⁢            β                                d            ⁢                                                  ⁢            t                              +                        N          p                ⁢        p              ;            E      ⁢                                    d            2                    ⁢          β                          d          ⁢                                          ⁢                      t            2                                =                            L          β                ⁢        β            -                        L          r                ⁢                              d            ⁢                                                  ⁢            β                                d            ⁢                                                  ⁢            t                              +                        L          p                ⁢        p            
This becomes a second order equation governing either roll rate or sideslip:
                    (                                                            N                p                            C                        ⁢                          E              A                                -                                    L              p                        A                          )            ⁢                                    d            2                    ⁢          β                          d          ⁢                                          ⁢                      t            2                                +                  (                                                            L                p                            A                        ⁢                                          N                r                            C                                -                                                    N                p                            C                        ⁢                                          L                r                            A                                      )            ⁢                        d          ⁢                                          ⁢          β                          d          ⁢                                          ⁢          t                      -                  (                                                            L                p                            A                        ⁢                                          N                β                            C                                -                                                    L                β                            A                        ⁢                                          N                p                            C                                      )            ⁢      β        =  0
The equation for roll rate is identical. But the roll angle, ϕ (phi) is given by:
            d      ⁢                          ⁢      ϕ              d      ⁢                          ⁢      t        =      p    .  
If p is a damped simple harmonic motion, so is ϕ, but the roll must be in quadrature with the roll rate, and hence also with the sideslip. The motion consists of oscillations in roll and yaw, with the roll motion lagging 90 degrees behind the yaw. The wing tips trace out elliptical paths. Note that in a rotating disk, yaw is continuous, and governed by the initial spin, rotational drag, and then slightly perturbed by other factors.
Stability
Stability requires the “stiffness” and “damping” terms to be positive. These are:
                                                        L              p                        A                    ⁢                                    N              r                        C                          -                                            N              p                        C                    ⁢                                    L              r                        A                                                                          N              p                        C                    ⁢                      E            A                          -                              L            p                    A                      ⁢                  ⁢          (      damping      )        ;                                                        L              β                        A                    ⁢                                    N              p                        C                          -                                            L              p                        A                    ⁢                                    N              β                        C                                                                          N              p                        C                    ⁢                      E            A                          -                              L            p                    A                      ⁢                  ⁢          (      stiffness      )      
The denominator is dominated by Lp, the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive.
Considering the “stiffness” term −LpNβ is positive because Lp is always negative and Nβ is positive by design. Lβ is usually negative, whilst Np is positive. Excessive dihedral can destabilize the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to Yβ.
The damping term is dominated by the product of the roll damping and the yaw damping derivatives, these are both negative, so their product is positive. The Dutch roll should therefore be damped.
The motion is accompanied by slight lateral motion of the center of gravity and a more “exact” analysis will introduce terms in Yβ etc. In view of the accuracy with which stability derivatives can be calculated, this is unnecessary.
Roll subsidence. Generating a control signal that causes an increase in roll rate and then ceasing the signal causes a net change in roll orientation. The roll motion with respect to aerodynamic factors is characterized by an absence of natural stability, there are no stability derivatives which generate moments in response to the inertial roll angle. A roll disturbance induces a roll rate which is only canceled by pilot or autopilot intervention. In a rotating disk, the gyroscopic action stabilizes roll angle independent of aerodynamics. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to:
            A      ⁢                        d          ⁢                                          ⁢          p                          d          ⁢                                          ⁢          t                      =                  L        p            ⁢      p        ⁢        
Lp is negative, so the roll rate will decay with time. The roll rate reduces to zero, but there is no direct control over the roll angle.
Spiral Mode.
Simply holding the stick still, when starting with the wings near level, an aircraft will usually have a tendency to gradually veer off to one side of the straight flightpath. This is the (slightly unstable) spiral mode.
Spiral mode trajectory In studying the trajectory, it is the direction of the velocity vector, rather than that of the body, which is of interest. The direction of the velocity vector when projected on to the horizontal will be called the track, denoted μ (mu). The body orientation is called the heading, denoted ψ (psi). The force equation of motion includes a component of weight:
            d      ⁢                          ⁢      μ              d      ⁢                          ⁢      t        =            Y              m        ⁢                                  ⁢        U              +                  g        U            ⁢      ϕ      
where g is the gravitational acceleration, and U is the speed.
Including the stability derivatives:
            d      ⁢                          ⁢      μ              d      ⁢                          ⁢      t        =                              Y          β                          m          ⁢                                          ⁢          U                    ⁢      β        +                            Y          r                          m          ⁢                                          ⁢          U                    ⁢      r        +                            Y          p                          m          ⁢                                          ⁢          U                    ⁢      p        +                  g        U            ⁢      ϕ      
Roll rates and yaw rates (in a rotating disk, the change in yaw rate is expected to be small) are expected to be small, so the contributions of Yr and Yp will be ignored.
The sideslip and roll rate vary gradually, so their time derivatives are ignored. The yaw and roll equations reduce to:
                              N          β                ⁢        β            +                        N          r                ⁢                              d            ⁢                                                  ⁢            μ                                d            ⁢                                                  ⁢            t                              +                        N          p                ⁢        p              =          0      ⁢                          ⁢              (        yaw        )              ;                              L          β                ⁢        β            +                        L          r                ⁢                              d            ⁢                                                  ⁢            μ                                d            ⁢                                                  ⁢            t                              +                        L          p                ⁢        p              =          0      ⁢                          ⁢                        (          roll          )                .            
Solving for β and p:
      β    =                            (                                                    L                r                            ⁢                              N                p                                      -                                          L                p                            ⁢                              N                r                                              )                          (                                                    L                p                            ⁢                              N                β                                      -                                          N                p                            ⁢                              N                β                                              )                    ⁢                        d          ⁢                                          ⁢          μ                          d          ⁢                                          ⁢          t                      ;      p    =                            (                                                    L                β                            ⁢                              N                r                                      -                                          L                r                            ⁢                              N                β                                              )                          (                                                    L                p                            ⁢                              N                β                                      -                                          N                p                            ⁢                              L                β                                              )                    ⁢                        d          ⁢                                          ⁢          μ                          d          ⁢                                          ⁢          t                    
Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle:
            d      ⁢                          ⁢      ϕ              d      ⁢                          ⁢      t        =      m    ⁢                  ⁢    g    ⁢                  (                                            L              β                        ⁢                          N              r                                -                                    N              β                        ⁢                          L              r                                      )                              m          ⁢                                          ⁢                      U            ⁡                          (                                                                    L                    p                                    ⁢                                      N                    β                                                  -                                                      N                    p                                    ⁢                                      L                    β                                                              )                                      -                              Y            β                    ⁡                      (                                                            L                  r                                ⁢                                  N                  p                                            -                                                L                  p                                ⁢                                  N                  r                                                      )                                ⁢    ϕ  
This is an exponential growth or decay, depending on whether the coefficient of ϕ is positive or negative. The denominator is usually negative, which requires LβNr>NβLr (both products are positive). This is in direct conflict with the Dutch roll stability requirement, and it is difficult to design a fixed wing aircraft for which both the Dutch roll and spiral mode are inherently stable.
Since the spiral mode has a long time constant, the pilot can intervene to effectively stabilize it, but an aircraft with an unstable Dutch roll would be difficult to fly. It is usual to design the fixed wing aircraft with a stable Dutch roll mode, but slightly unstable spiral mode.
Thus, for fixed wing aircraft, there is a well-developed mathematical model of flight dynamics. Other than the gyroscopic effects, the flight dynamics of a rotating disk will generally be less complex than a fixed wing aircraft, since yaw and change in yaw is largely irrelevant to the dynamics.
ROTATING AERODYNAMIC STRUCTURES. FIG. 1B defines various parameters of a spinning disk aerodynamic projectile. See, Scdary, A, “The Aerodynamics and Stability of Flying Discs”, Oct. 30, 2007, large.stanford.edu/courses/2007/ph210/scodary1/. To first approximation, a flying disc is simply an axi-symmetric wing with an elliptical cross-section. Of course, most ordinary wings would be unstable if simply thrown through the air, but the essential mechanics of its lift are mostly ordinary. The lift on a body is described by the lift equation, L=0.5CLAρV2, Where A is the cross-sectional area, p is the density, V is the free-stream velocity, and CL is the coefficient of lift, a semi-empirical constant that is a function of the shape of the object and its angle of attack with respect to the free-stream velocity of the fluid. Typically, CL increases linearly with the angle of attack, a, until some critical angle where the lift drops off steeply. If a typical commercial flying disc were a perfectly flat disc, CL would be zero at an angle of attack of zero. Although, the camber or shape of the disc allows for non-zero lift at an angle of attack zero. The coefficient of lift at zero angle of attack is often denoted CL0, and the increase of CL with α is denoted CLα. Hence, CL=CL0+CL&alphaα. Empirically, CL0 is roughly 0.188 and CLα is 2.37 for a typical flying disc. [1] This means an angle of attack of 15 degrees would have a coefficient of lift of roughly 0.62. While this value varies from disc to disc, it is to be expected that this quantity will be of order unity. The drag on the flying disc is defined in a fashion similar to that of the disk. The equation is now D=0.5CDAρV2. The only difference from the lift is that the coefficient of lift has been replaced by the analogous coefficient of drag. Unlike lift, drag typically depends on the angle of attack quadratically. So now, CD=CD0+CDαα2, and the empirical values have been found to be approximately CD0=0.15 and CDα=1.24. [1] The maximum ratio of lift to drag occurs around 15 degrees. For a wing with an elliptical cross-section, the center of pressure due to lift is offset ahead of the center of gravity.
Therefore, if one were to simply throw a flying disc, the lift would also cause a moment on the disc and cause it to flip over backwards. The key to the stability of the flying disc is its spin. The spin of the disc results in gyroscopic stability or pitch stiffness, and the greater the speed, the greater the stability. Typically, the moment due to lift and drag pressure on the disc is nearly perpendicular to the angular momentum of the spinning disc, and thus the disc experiences gyroscopic precession. The frequency of precession is Ω=M/Iω, where M is the moment, I is the moment of inertia of the disc about its axis of symmetry, and ω is the angular frequency of the disc's spin. This precession causes the disc to wobble as a gyroscope wobbles when its axis of spin is perturbed from the direction of gravity. Likewise, by spinning the disc, one trades roll stability for pitch stiffness. The equations of motion of the system, accounting for the external forces and moments is found to be [2] m (dv/dt+Ω×v)=F=mg and Idω/dt+Ω×Iω=M. The details of F and M depend on the aerodynamics of the disc. The typical mass of a flying disc is between 90 g and 175 g. The lighter discs maximize duration of flight, and the heavier discs will maximize distance thrown and minimize the effects of wind and stray currents. It is also apparent from the equations of motion that a greater moment of inertia I would also increase stability. The viscous no-slip condition at the boundary of the spinning disc causes the disc to generate some degree of vorticity. The circulation about the disc and the free-stream flow of air past the disc causes a force in the direction of the cross product of V with the angular momentum of the disc. This is attributed to the Magnus Effect, which is caused by one side of the disc having a higher free-stream velocity than the other, causing a pressure gradient. This causes a flying disc thrown clockwise to veer to the left, which is particularly noticeable as the viscous effects become more pronounced at the end of the flight.
Note that the center of pressure and the center of gravity are displaced. According to the present technology the center of gravity may be repositioned.
The disk rim at the edge of flying disc serves multiple important purposes. First and foremost, the thick rim eases gripping and tossing the disc. Without the thick rim, throwing a flying disc would be significantly more difficult. Additionally, the thick rim significantly increases the moment of inertia of the disc about the axis of symmetry, enhancing the stability of the disc. A flat plate without a thick rim, such as a dinner plate, has much less stability than a typical flying disc. Finally, the cupped region on the bottom of the disc substantially increases the coefficient of drag from the vertical profile, while the horizontal profile is still somewhat streamlined. As a result, as the disc begins to fall, the cupped profile behaves like a parachute, and the horizontal component of drag dwarfs the vertical component. This allows the flying disc to be thrown much further than a ball of equivalent velocity.
The top surface of a typical flying disk has several concentric grooves, which serve to trigger turbulence at the leading edge, which helps to keep the boundary layer of the flow over the top of the disc attached to the disc, substantially increasing lift. This effect also allows the disc to be thrown at a higher angle of attack before it stalls. A stall occurs when the flow separates from an object, causing a catastrophic decrease in lift and increase in drag.
One can cause a Frisbee to stall by simply throwing it at a very high angle of attack. The disc will quickly destabilize and fall to the ground. This permits an option of modulating the height of the ridges above the upper surface of the disk, to steer the disk. [1] S. Hummel and M. Hubbard, “Identification of Frisbee Aerodynamic Coefficients using Flight Data,” The International Conference on the Engineering of Sport, (Kyoto, Japan, Sep. 2002). [2] R. D. Lorenz, “Flight Dynamics Measurements on an Instrumented Frisbee,” Measurement Science and Technology, (2005) [3] J. R. Potts and W. J. Crowther, “Visualisation of the Flow Over a Disc-wing,” 9th Intl. Symp. on Flow Visualization, (2000). [3] J. R. Potts and W. J. Crowther, “Frisbee™ Aerodynamics,” 20th AIAA Applied Aerodynamics Conference and Exhibit, (St. Louis, Mo., 2002).
See, Lorenz, Ralph D. Spinning flight: dynamics of Frisbees, boomerangs, samaras, and skipping stones. Springer Science & Business Media, 2007, expressly incorporated herein by reference in its entirety. The spin axis of a rotating body precesses in a direction apparently orthogonal to the applied torque. The angular momentum of a rotating body is equal to the product of its moment of inertia (integral of mass times distance from axis of rotation) and its angular velocity. On the other hand, the rotational kinetic energy is equal to half of the product of the moment of inertia and the square of the angular velocity. Unless external moments are applied, the angular momentum of a system must remain constant. In general, the mechanics of rotation can be described by a set of expressions known as the Euler equations. The inertia properties can be represented by a tensor (a matrix of 9 numbers), but for most applications only three numbers (and often only two) are needed: Only the three diagonal terms in the tensor are nonzero. These are the moments of inertia about three orthogonal axes, the so-called principal axes. The principal axes are the axis of maximum moment of inertia, the axis of minimum moment of inertia, and the axis orthogonal to the other two. It can be shown that stable rotation only occurs about the minimum or maximum axis. Although it is possible to spin an object about its long axis, this motion is not necessarily stable in the long term. Specifically, internal energy losses force the object to ultimately rotate about the axis of maximum moment of inertia. However, internal energy dissipation (for example, flexing of imperfectly elastic elements or flow of viscous fluids) can absorb energy. If an object is spinning about an axis other than that of maximum moment of inertia, and energy is dissipated, the system must compensate for a drop in angular velocity by increasing the moment of inertia. In other words, it rotates such that the (constant) angular momentum vector becomes aligned with the maximum moment of inertia—the only way of reducing energy while keeping the angular momentum constant is to increase I as w decreases. The rotation about the axis of maximum moment of inertia, the stable end state, is sometimes called a “flat spin,” since the object sweeps out a flat plane as it rotates.
Precession is the movement of an angular momentum vector by the application of an external moment. Nutation is a conical motion due to the misalignment of the axis of maximum moment of inertia and the angular momentum vector. The maximum moment axis of the vehicle essentially rolls around in a cone around the angular momentum vector. Nutation is usually a very transient motion, since it is eliminated by energy dissipation. Coning refers to the apparent conical motion indicated by a sensor which is not aligned with a principal axis. Even in a perfectly steady rotation, a misaligned sensor will appear to indicate motion in another axis. A torque can be impulsive, i.e., of short duration. In such a case, the torque-time product yields an angular momentum increment which changes the direction of the vehicle's angular momentum before the body itself has had time to move accordingly. In this situation, the body spin axis (usually the axis of maximum moment of inertia) will be misaligned with the angular momentum vector. The vehicle will appear to wobble, due to the nutation motion. The amplitude of this wobble will decrease with time as energy dissipation realigns the spin axis with the new angular momentum vector. The rate of the wobble depends on the moments of inertia of the object: for a flat disk, where the axial moment of inertia is exactly double the transverse moments of inertia, the wobble period is half of the spin period.
Angular dynamics of aircraft are usually described by three motions: roll, pitch, and yaw. Roll denotes motion about a forward direction. Yaw is motion about a vertical axis, while pitch is motion in a plane containing the vertical and forward directions.
Aerodynamic forces and moments can be considered in several ways. Ultimately, all forces must be expressed through pressure normal to, and friction along, the surfaces of the vehicle. For most common flows, viscous forces are modest and only the pressures are significant. For a body to generate lift, pressures on its upper surfaces must in general be lower than those on its lower surface.
Another perspective is that the exertion of force on the flying object must manifest itself as an equal and opposite rate of change of momentum in the airstream. If an object is developing lift, it must therefore push the air down. Streamlines must therefore be tilted downwards by the object.
The distribution of pressures on the flying object will yield a resultant force that appears to act at an arbitrary position, the center of pressure. No torques about this point are generated. The weight (mg) of the vehicle acts at the center of mass (COM) (actually, the center of gravity, but since the flying object is typically exposed to a spatially uniform gravitational field, there is no meaningful difference) whereas the aerodynamic force acts at the center of pressure (COP) and is usually defined by a lift and drag, orthogonal and parallel to the velocity of the vehicle relative to the air. Because the COM and COP are not in the same place, there is a resultant pitch torque. The airflow hits the vehicle at an angle of attack.
The force is calculated as if it acted at the geometric center of the vehicle. Usually this force is expressed in three directions, referred to the direction of flight. Drag is along the negative direction of flight; lift is orthogonal to drag in the vehicle-referenced plane that is nominally upwards. The orthogonal triad is completed by a side-force. The forces are supplemented by a set of moments (roll, pitch, and yaw). These determine the stability of a vehicle in flight. Both forces and moments are normalized by dimensions to allow ready comparison of different sizes and shapes of vehicles. The normalization for forces is by the dynamic pressure (0.5 pV) and a reference area (usually the wing planform area). Dividing the force by these quantities, the residual is a force coefficient. These coefficients; dimensionless numbers usually with values of 0.001 to 2.0, are typically functions of Mach and Reynolds number (which are generally small and constant enough, respectively, to be considered invariant in the models) and of attitude.
The attitude (the orientation of the body axes in an external frame, e.g., up, north, east) may be compared with the velocity vector in that same frame to yield, in still air, the relative wind, i.e., the velocity of the air relative to the vehicle. In cases where there is a nonzero wind relative to the ground, an ambient wind vector may be added to the relative wind. In addition to changing the speed of the air relative to the vehicle, wind may be instrumental in changing the angle of the airflow relative to the body datum.
The most significant angle is that between the freestream and the body datum in the pitch plane. This is termed the angle of attack, and it is upon this parameter that most aerodynamic properties such as lift and moment coefficients display their most significant sensitivity. A second angle is relevant for conventional aircraft, and this is the angle of the freestream relative to body datum in the yaw plane: this is the sideslip angle. If a sideslip angle were to be defined relative to a body datum, it would vary rapidly owing to the body spin. The spin rate is usually not itself of intrinsic aerodynamic interest. However, when multiplied by a body length scale (the span of a boomerang, or the radius of a Frisbee) it corresponds to a tip or rim speed. This speed can be significant compared with the translational speed of the body's center of mass, and thus a measure of the relative speed is used, referred to as the advance ratio (cor/V).
Several parameters describe flight conditions: the density and viscosity of the fluid, the size and speed of the object, etc. These properties may be expressed as dimensionless numbers to indicate the ratio of different forces or scales. Because flow behaviors can be reproduced under different conditions but with the same dimensionless numbers, these numbers are often termed similarity parameters. The Mach number, for example, is simply the ratio of the flight speed to the speed of sound in the medium. Since the sound speed is the rate of propagation of pressure disturbances (=information), a Mach number in excess of 1 indicates a supersonic situation where the upstream fluid is unaware of the imminent arrival of the flying object, and the flow characteristics are very different from subsonic conditions. In particular, a shock wave forms across which there is a discontinuous jump in pressure and temperature as the flow is decelerated. This shock wave typically forms a triangle or (Mach) cone around the vehicle, with a half angle equal to the arctangent of the Mach number. The objects considered herein remain within a subsonic regime, and Mach number variations (<<1) are not of concern. Much more important is the Reynolds number, the ratio of viscous to inertial (dynamic pressure) forces. This may be written Re=vlρ/μ, where v is the flight speed, I a characteristic dimension (usually diameter, or perhaps a wing chord), μ the (dynamic) viscosity of the fluid, and p the density of the fluid. The dynamic viscosity is the ratio of the shear stress to the velocity gradient in a fluid. This is a constant for a given fluid, and is what is measured directly. The symbol for this property is usually Greek mu, and the units are those given by stress/velocity gradient=Newtons per meter squared, divided by meters per second per meter, thus Newtons per meter squared, times second, or Pascal-seconds (Pa-s) in SI units, or the Poise (P). 1 P=10 Pa-s.
The effects of Reynolds number cause variation of drag coefficient. At very low values of Re (<1), viscous forces dominate, and the drag coefficient for a sphere is equal to 24/Re, which may be very large. Since the drag coefficient Ca is defined with respect to the fluid density, which directly relates to the (insignificant) inertial forces, this relation for Ca is equivalent to substituting a formula for drag that ignores density and instead relates the drag only to viscosity. This substitution leads to Stokes' law for the fall velocity of spheres in a viscous fluid. As the Reynolds number increases (the flow becomes “faster”), the inertial forces due to the mass density of the fluid play a bigger and bigger role. The flow becomes unable to stick to the back of the sphere, and separates. At first (Re of a few tens) the flow separates from alternate sides, forming two lines of contra-rotating vortices. This is sometimes called a von Karman vortex street. This alternating vortex shedding is responsible for some periodic flow-driven phenomena such as the singing of telephone wires. Control of drag is essentially equivalent to controlling the wake—whatever momentum is abstracted from the flow onto the object (or vice versa) is manifested in the velocity difference between the wake and the undisturbed fluid. If the wake is made more narrow, then the momentum dumped into it, and thus the drag, will be kept small One way of doing this is by streamlining. Another circumstance is to make the boundary layer (the flow immediately adjacent to the object) turbulent, which allows it to “stick” better to the object and thus make the wake more narrow.
Symmetric control of the boundary layer is of course known in golf, whereby the dimples on the ball increase the surface roughness so as to make the boundary layer everywhere turbulent. The turbulent boundary layer is better able to resist the adverse pressure gradient on the lee (downstream) side of the ball and remains attached longer than would the laminar layer on a smooth ball. The result is that the wake is narrower and so drag is lessened.
Similar boundary layer control is sometimes encountered on other (usually cylindrical) structures that encounter flow at similar Reynolds numbers. An example is the bottom bar on a hang glider. This bar can be faired with an aerofoil, but such a shape is harder to grip with the hands, and a cylindrical tube is rather cheaper. However, a smooth cylinder has a high drag coefficient, so a “turbulating” wire is often attached at the leading side to trip the boundary layer into turbulence and so reduce drag.
It is often said that the airflow across the curved top of a wing is faster than across the bottom, and since Bernoulli's theorem states that the sum of the static pressure and dynamic pressure in a flow are constant, then the faster-flowing (higher dynamic pressure) air on the upper surface must “suck” the wing upwards. This is sometimes true in a sense, but it is a rather misleading description—it fails utterly, for example, to explain why a flat or cambered plate can develop lift. In these cases in particular, the air travels the same distance over top and bottom, and so faster flowing air is not required on the upper surface, at least not from geometric considerations alone. Conservation of momentum dictates that if the airflow is to exert a lift force on the wing, then the wing must exert a downward force on the air. The flow of air past the wing must be diverted downwards. Whatever causes the lift, a result will be a downwards component of velocity imparted to the air. In an idealized sense, one can imagine the downward diversion of the airflow as a rotation of the streamlines, and consideration of the wing as a circulation-inducing device is a powerful idea in fluid mechanics.
As for pressure, which is force per unit area, there will indeed be on average a lower pressure on the upper surface of a lifting wing than on the lower surface. The net force is simply the integral of the pressure over the wing. Indeed, since both of these aspects of the flow will depend on the shape of the wing, its orientation (a flat plate inclined slightly upwards to the flow will obviously divert the flow downwards), and on how the flow stays attached to the wing, one might consider both the flowfield and the pressure field to be effects. But it is not always true that the air flows faster across the upper surface. The attachment of the flow is crucial. Once the flow has passed the suction peak, the area on the upper surface of the wing where pressure is least, the boundary layer may struggle to remain attached. If the layer separates, the drag will increase (as for a smooth sphere) and the lift will be reduced. This condition is known as the “stall.”
The Magnus effect (or Robbins-Magnus effect) provides that, even a perfectly symmetric sphere can be made to veer in flight by causing it to spin. The airflow over the side of a ball (or cylinder) that is spinning against the direction of fluid flow will cause the flow to separate earlier, while the flow running with the rotating surface will stick longer. The result is that the wake is diverted sideways. Because the flow is diverted sideways, there must be a reaction on the spinning object, a side-force. This side-force causes spinning objects to veer in flight, a ball “rolling forward” (i.e., with topspin) will tend to swing down, while a ball with backspin will tend to be lofted upwards, a key effect in golf. In essence the ball “follows its nose.” If the spin axis is vertical, then the ball is diverted sideways. The Magnus force is explained just as lift and drag-momentum dumped into the wake via control of flow separation.
See, Magnus, G., Ann. Phys. Chem. 88, 1-14, 1953; Rizzo, F. The Flettner rotor ship in the light of the Kutta-Joukowski theory and of experimental results, NACATN 228, October 1925; Robins, B., New Principles of Gunnery Containing the Determination of the Force of Gunpowder and Investigation of the Difference in the ResLiting Power of the Air to Swift and Slow Motion, 1742; Talay, T., Introduction to the aerodynamics of flight, NASA SP-367, 1975. Tapan K. Sengupta and Srikanth B. Talla, Robins-Magnus effect: A continuing saga, Current Science Vol. 86, No. 7, 10 Apr. 2004; Tokaty, G. A., A History and Philosophy of Fluid Mechanics, Dover, 1994, expressly incorporated herein by reference.
Some balls have peculiar dynamics owing to nonspherical shape, whereas others are round but exploit various aspects of boundary layer control. These aspects include overall surface roughness, such as the dimples of a golf ball, and asymmetric roughness, such as the seam of a cricket ball or baseball. Still other complications are introduced by spin, which may generate lift via the Robins-Magnus effect.
For the most part, the size range of balls used in sports is limited to the range that can be conveniently manipulated and thrown with a single hand. This spans the range from a couple of centimeters diameter (squash balls, golf balls, etc.), where the ball is placed or thrown into play single-handedly, to larger balls (−25 cm) such as footballs and basketballs which can, with skill, be thrown from a single open hand, but are often thrown with two. The speed range is in general that which can be achieved by the unaided human arm; a few tens of meters per second, with the peak speed falling with ball mass once the ball adds significant inertia to the arm-ball system. Particular fast-movers are cricket balls and baseballs, which at ˜150 g are perhaps an optimum mass for throwing. Golf balls fly particularly fast because of the large moment arm of the club used to strike them, and the elastic properties of club and ball. Footballs (soccer balls) generally fly faster than other balls of the same size (such as basketballs) because they are kicked, the long moment arm of leg, and the large mass of the foot, make an efficient club. The key property affecting a ball's general flight is its mass:area ratio. Squash balls and table-tennis balls have the same size, but the latter has a much lower mass. In principle, one could throw a table-tennis ball a little faster than a squash ball, but one would be unlikely to throw it further, since air drag has a much stronger effect on the table-tennis ball. At a given speed, the drag force on the two balls is the same, but the change in flight speed (i.e., the acceleration, equaling force/mass) is much higher for the table-tennis ball.
The other aspect affecting a ball's drag (without spin or asymmetric boundary layer control, a ball has no lift) is the flight Reynolds number. The drag coefficient of spheres is somewhat constant over a wide range of Reynolds number, but falls appreciably at a critical Reynolds number which ranges between about 4×104 to 4×105, depending on surface roughness. It can be seen that the golf ball, which has a relatively high surface roughness, has a drag coefficient that drops at a low Reynolds number. The somewhat smoother soccer ball has a drop at a higher Reynolds number, but because the ball itself has a larger diameter, this critical Reynolds number corresponds to a lower flight speed than the golf ball. A perfectly smooth sphere has the highest critical Reynolds number.
The texture of a tennis racquet is such that the effective coefficient of friction between the racquet and ball is quite large, permitting very large spin rates to be induced; indeed, regulations exist on the stringing style of a racquet to restrict the amount of spin. The spin permits large side-forces (i.e., “lift”) to be developed by the Magnus force. A tennis ball flies at speeds well above the critical Reynolds number, and thus the boundary layer is tripped into turbulence with a spin-dependent location. The usual application is in the “topspin lob” whereby a topspin causes the ball to dive downwards, permitting a fast shot that still hits the court within the permitted boundary.
The game of golf involves some of the highest speed motion in ball sport—a small ball is struck with a massive club, which provides the largest possible moment arm, the length of the lever (arm+club) being dictated by the distance between the shoulder and the ground. The shape of a gold club is designed not only to provide a translational velocity impulse, but also to impart a backspin to provide lift. The angle of the club head, its shape, and the presence of grooves to increase friction all play a role. Wind tunnel tests by Bearman and Harvey in 1949 using a motorized assembly to spin the ball show that the classic golf ball, at a Reynolds number of 105, has a lift coefficient that varies from 0 with zero spin rate to about 0.3 when the advance ratio (the circumferential speed divided by the flight speed) reaches about 0.4, corresponding to about 6000 rpm. The drag coefficient rises from about 0.27 to 0.34 over the same range of advance ratio. The lift coefficient for a given spin rate is highest at low speeds, since the advance ratio is correspondingly high, although the variation is small above speeds of 55 m/s. Populating the surface of a sphere with a regular pattern without a preferred orientation is not a trivial problem, and thus not only the number or density of the dimples, but also their arrangement on the ball surface influences performance in play.
The patches of a soccer ball are stitched together to form a truncated icosahedron (i.e., one of the perfect solid shapes known to the ancient Greeks, made only of 20 isosceles triangles, but with some corners chopped off). The resultant shape is one with 60 corners, 12 pentagonal faces (often painted black), 20 hexagonal ones (white), and 90 edges. Carre et al. (2002) used video measurements (240 frames per second) of the trajectory of a kicked football (A Mitre Ultimax, diameter 215 mm, mass 415 g) to infer lift and drag coefficients. Their drag coefficient for nonspinning balls is a surprisingly strong function of kick speed, varying from about 0.05 at 20 m/s to 0.35 at 30 m/s, increasing rather than decreasing with Reynolds number. Perhaps some deformation of the ball occurs in hard kicks. The lift coefficient was close to zero for a nonspinning ball (as the broadly symmetric pattern of seams would suggest); however, as the spin rate increased to 50-100 radians per second, the lift coefficient rose to about 0.25, and was more or less constant for higher spin rates. Here, of course, the lift is due to the Robins-Magnus effect. Based on these data, they note that a football kicked at 18 m/s at an angle of 24 degrees from horizontal could fly at a range of 10 m anywhere between 1.2 and 3.2 m from the ground, purely by varying the spin rate at launch; topspin of course leading to lower altitude and backspin providing lift upwards. The most common deliberate application of this side-force is in the corner kick, and in a free kick when a player attempts to bend the ball around a wall of defenders.
A cricket ball has an equatorial plane with a set of stitching along it. This stitching acts as a boundary layer control structure; if the equator is held (by spin) at an angle to the airflow, then the flow on one half will encounter the stitches (and thereby have its boundary layer tripped into turbulence) on the leading hemisphere, while the other half will not encounter it until later. The side that encounters the stitching may transition into turbulence, and thus is able to “stick” to the ball surface better through the adverse pressure gradient on the trailing side. In the absence of spin, the seam would slowly rotate around and average the side-force down to nothing. Thus, spin is applied to maintain a constant orientation of the seam, rather than to develop aerodynamic forces per se through the Magnus-Robins effect. The optimum side-force (side-force coefficient CF˜0.3) is achieved when the seam equator makes an angle of about 20 degrees with the oncoming airflow. Similar results can be obtained with the seam at zero degrees if one hemisphere of the ball is smooth and the other rough. During play, bowlers allow one side of the ball to become rough, while rubbing (sometimes augmented with sweat or saliva) keeps one side smooth. Overt roughening is forbidden. An interesting change of behavior occurs at high speed. While a “swing” that amounts to almost 1 m in the “pitch” length of 20m is possible at moderate speeds, such trajectories are only possible for a narrow range of throw speeds. The generation of a side-force depends on the differential separation of the boundary layer one side must separate before the other. If the boundary layer on both sides were strongly turbulent, such that both separate at more or less the same place, then the seam would make little difference. This situation occurs in fast bowls; the ball can be thrown at up to 40 m/s (Re˜1.9×105). Because the boundary layer becomes naturally turbulent even in the absence of the seam at Re˜1.5×105, the side-force coefficient begins to fall off at this speed (˜30 m/s).
A baseball's seam of over 200 stitches joins together two hourglass-shaped strips of leather. Although the ball is of a broadly similar size (9-9¼ inches, circumference) and mass (5-5¼ ounces avoirdupois) as a cricket ball, its motions are more complex since the stitching is not so simply arranged. There are two principal pitches in baseball of aerodynamic interest, the curveball and the knuckleball. (Certain other pitches are named, such as the screwball, which is essentially a curveball with the spin axis reversed, and the slider, a fast pitch with the spin axis vertical. All are, in essence, just variants of a curveball; no different aerodynamic effects are invoked.) A less common pitch, perhaps, is the knuckleball, which is somewhat related to the swing bowl in cricket, in that the seam is used to trip turbulence asymmetrically. However, here the configuration of the seam does not permit a constant orientation by spinning. Instead, the pitcher attempts to throw with as little spin as possible. There is inevitably some rotation, which has the effect of causing the seam to be presented at a range of angles to the flow and thus cause a varying, and therefore hard for the batter to anticipate, side-force direction. It is estimated that a knuckleball may deviate by 27 cm from its initial trajectory before returning.
A softball is somewhat larger than a baseball and will typically be thrown more slowly (underhand pitch). Its size relative to the hand makes it much harder to impart spin to the ball. Thus, although slower pitches give longer times for side-forces to act, and low flight speeds give higher advance ratios for a given spin, in general spin effects on softballs are rather modest.
The table-tennis ball is the light and smooth. Aerodynamically it resembles a squash ball in size and smoothness, but has a mass/area ratio 10 times lower. It thus decelerates rapidly due to air. It is also more responsive to other aerodynamic forces such as the Robins-Magnus effect. Like a table-tennis ball, a squash ball is smooth. However, a squash ball has a much higher mass/area ratio, and thus its trajectory is less affected by aerodynamic forces, and is in fact nearly ballistic. The art of the game derives mostly from the kinematics of the bounce from the walls; use of multiple bounces rapidly eliminates the ball's kinetic energy. Topspin is used to cause a ball to bounce steeply downwards, making it hard to intercept and return in time, while backspin is used in the serve, to cause an upward bounce making for a steep descent at the back of the court. Squash is notable in that the game's thermal component is very obvious; the coefficient of restitution of the ball is highly dependent upon the ball's temperature. Two effects are at work: the elasticity of the rubber, and the pressure of the air inside. A remarkable feature of a sliced (i.e., highly spun) ball in squash is that the ball becomes visibly distorted.
The stretched prolate shape of balls used in American football and in rugby introduce complications. The projected area of the ball is almost halved when the ball is end-on compared to when it is flying broadside. Thus, as is well known, the ball can be thrown further if thrown longwise. The ball is not statically stable (at least not appreciably so) and thus a spin must be imparted to the ball in order to keep its long axis pointed in the direction of motion. Thus most good passes are “spiral” in nature. Video and flight data recorder measurements suggest 600 rpm is a typical spin rate for a thrown football. Because of the tendency for aerodynamic moments to precess the ball, and the intrinsic instability of rotation around the axis of minimum moment of inertia, the ball tends to begin nutating, if not tumbling, towards the end of the throw. Some of the ball's angular momentum is shed into the wake, which would ultimately cause the spin to decrease. The axial acceleration of a football is somewhat constant, declining as the ball slows in flight (and perhaps also because the axis begins to cone around the direction of flight). The transverse acceleration is modulated by the spin (−6 revolutions per second, slowing towards the end of flight). The moments of inertia of a football are such that this tumbling, as measured by in-flight accelerometer measurements (Nowak et al., 2003), has a period 1.8 times the initial spin period. The tumbling (or rather, nutation) period is longer than the spin period for long objects; for flat discs the nutation period is close to half of the spin period.
Many balls operate at or near a critical Reynolds number, such that drag coefficient can be a strong function of flight speed. Substantial side- or lift-forces are developed by many balls via seam-triggered boundary layer transition and thus delayed separation; in other cases, Robins-Magnus lift plays the dominant role. See, Adair, R., The Physics of Baseball, 3rd Ed. HarperCollins, 2002; Bahill, A. T., D. G. Baldwin, and J. Venkateswaran, Predicting a baseball's path, American Scientist, 93(3), 218-225, May-June 2005; Barkla, H., and A. Auchterlonie, The Magnus or Robins effect on rotating spheres, Journal of Fluid Mechanics, 47, 437-447, 1971; Bearman, P. W., and J. K. Harvey, Golf ball aerodynamics, Aeronauts Quarterly, 27, 112, 1976; Carre, M. J., T. Asaum, T. Akatsuka, and S. J. Haake, The curve kick of a football II: Flight through the air, Sports Engineering 5, 193-200, 2002. Cooke, Alison J., An overview of tennis ball aerodynamics, Sports Engineering, 3, 123-129, 2000; Mehta, R. D., Aerodynamics of sports balls, Annual Reviews of Fluid Mechanics, 17, 151-189, 1985; Nowak, Chris J., Venkat Krovi, William J. Rae, Flight data recorder for an American football, Proceedings of the 5th International Conference on the Engineering of Sport, Davis, Calif., Sep. 13-16, 2004; Rae, W. J., and R. J. Streit, Wind tunnel measurements of the aerodynamic loads on an American football, Sports Engineering, 5, 165-172, 2002. Rae, W. J., Flight dynamics of an American football in a forward pass, Sports Engineering, 6, 149-164, 2003; Lord Rayleigh, On the irregular flight of the tennis ball, Mathematical Messenger 7, 14-16, 1878; Stepanek, Antonin, The aerodynamics of tennis balls: The topspin lob, American Journal of Physics 56, 138-142, 1988; Tait, P. G., Some points in the physics of golf, Nature 42, 420-423, 1890. Continued in 44, 497-498, 1891 and 48, 202-204, 1893; Thomson, J. J., The dynamics of a golf ball, Nature 85, 251-257, 1910 Volume 3, Issue 2, 123, May 2000; Watts, R. G., and G. Moore, The drag force on an American football, Am. J. Phys, 71, 791-793, 2003; Watts, R. G., and A. T. Bahill, Keep Your Eye on the Ball, Freeman, N.Y., 2000; www.hq.nasa.gov/alsj/a14/a14.html; news.bbc.co.uklsportacademy/hi/sa/tennis/features/newsid_2997000/2997504.stm, expressly incorporated herein by reference.
A flat plate tends to pitch up in flight, and this tendency must be suppressed in order to have sustained flight. The basics are outlined in Schuurman (1990) and Lorenz (2004). This suppression is achieved by some combination of aerodynamic tuning to reduce the pitch-up moment and the application of spin to give gyroscopic stiffness. These are, however, only palliative measures that extend the duration of level flight-simple adjustment of shape and flight parameters cannot keep an object flying forever for the following reason: Spin stabilization only slows the destabilizing precession due to the pitch-up moment—the useful flight time is only a transient interval whose duration is proportional to the spin rate divided by the pitch moment. Of course, if the pitch moment could be made zero, then the spin axis precession would take an infinitely long time. However, a static design disk flying shape does not have a zero pitch moment at all angles of attack, and since the angle of attack will change in flight due to the changing flight speed and flight path angle (due to the actions of gravity, lift, and drag), then sooner or later the pitch moment must be dealt with. The “Professional Model Frisbee Disc” received the U.S. Pat. No. 3,359,678 in 1964.
Several s sports have developed using the Frisbee®, notably Guts, Freestyle, Ultimate, and Disc Golf.
Guts is a game in which two opposing teams take turns throwing the disc at each other, the goal being to have the disc hit the ground in a designated zone without being caught. Freestyle is more of a demonstration sport like gymnastics, with exotic and contorted throws, catches, and juggles evaluated for difficulty, precision, and artistic impression.
Ultimate is a team passing game with similarities to basketball and American football. Teammates work the Frisbee forward across a 70-yard long field, 40 yards across, by passing from one team member to the other. If possession is lost, either by the disc going out of bounds, falling to the ground without being caught, or being caught by a member of the opposing team, the opposing team takes possession. A point is scored when the disc is caught in the end-zone.
A thrower in Ultimate must toss to a teammate while avoiding interception, and therefore curved flights are essential. Sometimes the thrower may be blocked by an opposing player and thus must use an exotic throw, such as the overhead “hammer”, where the disc is thrown over the shoulder in a vertical orientation, to roll onto its back and fly at near −90 degrees angle of attack. The catcher must anticipate how long the disc may hang in the air, and especially any turns it may make towards the end of its flight.
The Frisbee may also be used as a target sport. In one variant, a golf-like game, the aim is to hit the targets in as few throws as possible. Headrick patented a Disc Pole Hole, which could catch a Frisbee. The device consists of a frame supported by a pole: ten chains hang down from the frame, forming a paraboloid of revolution. This paraboloid sits above a wide basket. The chains absorb the momentum of a correctly thrown disc and allow it to fall into the basket (without the chains, a disc would typically bounce off the pole, making scoring near-impossible).
“Hot Zone,” is a competition sport where a player throws a Frisbee to be caught in a specified zone by a dog (often a sheepdog breed).
A Frisbee is a low-aspect ratio wing—in that sense its lift and drag can be considered conventionally. In common with many low-aspect ratio shapes, its behavior may be predictable. The key is the pitch moment and how to mitigate its effects. The conventional Frisbee does this in two ways. First, the deep lip reduces the pitch moment to manageable values. Secondly, the thickness of the plastic in a Frisbee is adjusted across the disc, such that much of the disc's mass is concentrated at the edge, to make a thick lip. This has the effect of maximizing the moment of inertia, making the Frisbee like a flywheel. The precession rate is equal to the pitch moment divided by the moment of inertia and spin angular velocity. Keeping the precession down to a few degrees over the flight duration of a couple of seconds is important to stable flight. (According to one aspect of the technology, the pitch moment may be varied, for example by altering a radial displacement of a set of balanced masses or fluidic mass. The aerodynamics may also be controlled during flight to alter the pitch moment).
Nakamura and Fukamachi (1991) performed smoke flow-visualization experiments on a Frisbee at low flow velocity in a wind tunnel (1 m/s). The disc (a conventional, although small, Frisbee) was spun with a motor at up to 3 times per second, yielding an advance ratio of up to 2.26. The smoke indicated the presence of a pair of downstream longitudinal vortices (just like those behind a conventional aircraft) which create a downwash and thus a lift force. These investigators also perceived an asymmetry in the vortex pattern due to the disc's spin, and also suggested that the disc spin increased the intensity of the down-wash (implying that the lift force may be augmented by spin). The low flow velocity (20 times smaller than typical flights) used in these experiments may have given a disproportionate effect of rotation.
Yasuda (1999) measured lift and drag coefficients of a flying disc for various flow speeds and spin rates. He additionally performed a few free-flight measurements on the disc (with the disc flying a short distance indoors in the field of view of a camera) and wind tunnel measurements on a flat disc. His free-flight measurements on a conventional disc show that typical flight speeds are 8-13 m/s and rotation rates of 300-600 rpm (5-10 revolutions per second) and the angle of attack was typically 5-20 degrees. The most common values for these parameters were about 10.5 m/s, 400 rpm, and 10 degrees, respectively. The flat disc had a zero lift coefficient at zero angle of attack, and a lift curve slope between 0 and 25 degrees of 0.8/25. The Frisbee had a slight lift (CL−0.1) at zero angle of attack, and a lift curve slope of −1/25. The Frisbee paid a price for its higher lift: its drag was commensurately higher. The flat plate had a drag coefficient at zero angle of attack of 0.03 and at 25 degrees of 0.4; the corresponding figures for the Frisbee were 0.1 and 0.55. (The drag curves are parabolic, as might be expected for a fixed skin friction drag to which an induced drag proportional to the square of the lift coefficient is added.) Yasuda notes that the lift: drag ratio of a flat plate is superior to that of the Frisbee. No significant dependence of these coefficients on rotation rate between 300 and 600 rpm was noted.
Higuchi et al. (2000) performed smoke wire flow visualization and PIV (particle image velocimetry) measurements, together with oil flow measurements of flow attachment on the disc surface. They used a cambered golf disc, with and without rotation and (for the most part) a representative flight speed of 8 m/s, and studied the downstream vortex structure and flow attachment in some detail.
To date, the most comprehensive series of experiments have been conducted by Jonathan Potts and William Crowther at Manchester University in the UK, as part of the Ph.D. research of the former. One aim of the research was to explore the possibilities of control surfaces on a disc wing. In addition to measuring lift, drag, and pitch moment at zero spin for the classic Frisbee shape, a flat plate, and an intermediate shape, these workers measured these coefficients as well as side-force and roll moment coefficients for the Frisbee shape at a range of angles of attack and spin rates. These coefficients will be discussed in a later section. Additionally, Potts and Crowther performed pressure distribution measurements on a nonspinning disc, smoke wire flow visualization, and oil flow surface stress visualizations. (They performed these on the regular Frisbee shape, and one with candidate control surfaces.)
Potts states, “The simulation is also used to demonstrate that with control moments from suitable control effectors, it is possible to generate a number of proscribed flight manoeuvres”; also “Finally, it has been shown that with appropriate initial conditions and appropriate control moment input, it is feasible to explore hypothetical disc-wing manoeuvres such as a spiral turn and a spiral roll.” The model of flight is developed in Section 7.3.4 of the Potts dissertation. Potts, Jonathan R., and William J. Crowther. “Flight control of a spin stabilised axi-symmetric disc-wing.” 39th Aerospace Sciences Meeting, Reno, Nev., USA. 2001, discloses various flight control methods, none of which are spin-angle synchronized. (“The control forces generated by a disc-wing with installed turbulence strips are capable of producing a banked turn manoeuvre of around a 100 m radius for the rotating case. If this degree of aerodynamic control could be gained from active on/off turbulence strips or some other method of control than that would offer practical possibilities for aerodynamic control of a disc-wing UAV”).
A useful and instructive comparison can be made between a flat plate and a Frisbee. Let us first consider drag. The drag coefficient is the drag force normalized with respect to dynamic pressure (0.5 ρV2) and the planform area of the disc. Since at low incidence angles the area of the disc projected into the direction of flow is very small (they used a plate with a thickness:chord ratio of 0.01), it follows that a flat plate will have a very low drag coefficient, ˜0.02. On the other hand, the Frisbee, with its deep lip (thickness:chord ratio of 0.14) has a much larger area projected into the flow, and its drag coefficient at zero angle of attack is therefore considerably larger (˜0.1). The Frisbee maintains a more or less constant offset of 0.1 above the value for a flat plate. This in turn has a parabolic form with respect to angle of attack, owing to the combination of a more or less constant skin friction drag term and the induced drag term, which is proportional to the square of lift coefficient. While a flat plate has zero lift at zero angle of attack, and a lift coefficient that increases with a slope of ˜0.05/degree, the Frisbee, having a cambered shape, develops appreciable lift at zero angle of attack (CL0˜0.3), its lift curve slope is similar.
The major difference between the Frisbee and flat plate is in the pitch moment coefficient. While this is zero for a flat plate at zero angle of attack (which is not a useful flying condition, since a flat plat develops no lift at this angle), it rises steeply to ˜0.12 at 10 degrees. Because the Frisbee's trailing lip “catches” the underside airflow and tries to flip the disk forward, the pitch-up tendency of the lift-producing suction on the leading half of the upper surface is largely compensated. Its pitch moment coefficient is slightly negative at low incidence and is zero (i.e., the disc flies in a trimmed condition) at an angle of attack of about 8 degrees. Over the large range of angle of attack of −10 to +15 degrees, the coefficient varies only between −0.02 and +0.02.
The existence of a trimmed position (pitch moment coefficient CM=0) permits the possibility of a stable glide. If the disc is flying downwards at a speed (dictated by the lift coefficient at the trimmed condition) such that drag is balanced by the forward component of weight, then the speed will remain constant. However, although the zero pitch moment means the disc will not roll, the roll moment is not zero, and so the spin axis will be slowly precessed forward or back, changing the angle of attack.
Hummel has pointed out the role of the sign change in pitch moment in causing the sometimes serpentine (S-shaped) flight of Frisbees. When thrown fast at low angle of attack, the pitch moment is slightly negative and causes the Frisbee to very slowly veer to the right. However, as the disc's speed falls off, its lift no longer balances weight, and it falls faster downwards, increasing the angle of attack. When the angle of attack has increased beyond 9 degrees, the pitch moment becomes positive and increases rapidly. This leads to the often-observed left curve at the end of a flight.
Potts and Crowther also study the side-force coefficient (which might be thought of as due to the Robins-Magnus force, although in reality it is rather more complicated, since most of the boundary layer develops over the flat surface of the disk, rather than its somewhat cylindrical lip) and the roll moment. The side-force coefficient is not strongly variable over the range of angles of attack studied (−5 to 15 degrees). It does vary with spin rate. For low values of advance ratio AR (<0.5, at an airspeed of 20 m/s) the coefficient is just slightly positive (0.02). However, for more rapid spin, the coefficient increases, at AR=0.69, Cs=0.04-0.05, and for AR=1.04, Cs˜0.8. To first order, then, these data show that the side-force coefficient is proportional to advance ratio; a reasonable expectation is that the coefficient is in fact directly proportional to the tip speed, although this parameter was not varied independently in these tests. Although the lift and drag coefficients were not appreciably affected by spin, the pitch moment did become more negative (by 0.01, almost a doubling) at 0-10 degrees angle of attack as the spin rate was increased from AR=0 to 1. The roll moment coefficient was also determined, and was almost zero (within 0.002 of zero) for low spin rates and more or less constant with angle of attack over the range −5 to 15 degrees.
However, the higher aspect ratio data showed a significant roll moment Cm˜−0.006 for advance ratio AR=0.7 and CM˜−0.012 for AR=1: in both cases the moment coefficient increased in value with a slope of about 0.0006/degree.
Studies have revealed the prominent existence of nutation in the early part of the throw. A good throw will avoid exciting nutation, which seems to substantially increase drag. It can be seen in some photographs of hard Frisbee throws that the disc becomes visibly deformed by inertial loads. The disc is held only at one edge, and to reach flight speeds of ˜20 m/s in a stroke of only a meter or so requires ˜20 g of acceleration, which for the half disc (90 grams) being accelerated at this rate corresponding to a force of 20 N; equivalent to hanging a ˜2 kg weight at the edge of the disc, and causing a transient deformation that might excite nutation.
In-flight measurements offer the prospect of measuring flow properties such as pressure on the rotating disc. Pressure distribution measurements are confounded by radial forces of the rotating disk, while microphones show how as the angle of attack increases, the pressure fluctuations on the disc become larger even as the flight speed decreases towards the end of the flight.
Lorenz (Lorenz, Ralph D. Spinning Flight: Dynamics Of Frisbees, Boomerangs, Samaras, And Skipping Stones. Springer Science & Business Media, 2007, P. 189) states: “A control surface, such as a flap, would have little useful effect on a Frisbee's flight were it to be simply fixed onto the disc. As the disc spins around, the control effect would vary or even reverse, and the spin-averaged effect would be small. However, if on-board sensors were used to trigger a fast-acting flap at a particular phase of rotation, the prospect of a maneuverable Frisbee can be envisaged. This might simply involve some stability augmentation, say to suppress the hook at the end of the flight. But much more appealing ideas become possible, a Frisbee with a heat sensor to detect a player, such that the disc tries to avoid being caught!”
The challenge in the Frisbee throw is that the overall flight is very sensitive to the initial parameters; small variations in angle of attack can lead to very different flights.
See, Higuchi, H., Goto, Y., Hiramoto, R., & Meisel, I., Rotating flying disks and formation of trailing vortices, AMA 2000-4001, 18th AIAA Applied Aero. Conf., Denver, Colo., August 2000; Hubbard, M., Hummel, S. A., 2000. Simulation of Frisbee flight. In Proceedings of the 5th Conference on Mathematics and Computers in Sports, University of Technology, Sydney, Australia; Hummel, S., Frisbee Flight Simulation and Throw Bionwchanics, M.Sc. thesis, UC Davis, 2003; Johnson, S., Frisbee—A Practitioner's Manual and Definitive Treatise, Workman Publishing, 1975; Lorenz, R. D., Flight and attitude dynamics of an instrumented Frisbee, Measurement Science and Technology 16, 738-748, 2005; Lorenz, R. D., Flight of the Frisbee, Engine, April 2005b; Lorenz, R. D., Flying saucers, New Scientist, 40-41, 19 Jun. 2004; Malafronte, V., The Complete Book of Frisbee: The History of the Sport and the First Official Price Guide, American Trends Publishing, 1998; Nakamura Y. & Fukamachi N., Visualisation of flow past a Frisbee, Fluid Dyn. Res., V7, pp. 31-35, 1991; Potts, J. R., & Crowther, W. J., Visualisation of the flow over a disc-wing. Proc. of the Ninth International Symposium on Flow Visualization, Edinburgh, Scotland, UK, August 2000; Potts, J. R., Crowther, W. J., 2002. Frisbee aerodynamics AIAA paper 2002-3150. In Proceedings of the 20th AMA Applied Aerodynamics Conference, St. Louis, Mo.; Rohde, A. A., Computational Study of Flow around a Rotating Disc in Flight, Aerospace Engineering Ph.D. dissertation, Florida Institute of Technology, Melbourne, Fla., December 2000; Schuurmans, M., Flight of the Frisbee, New Scientist, July 28, 127 (1727) (1990), 37-40; Stilley, G. D., & Carstens, D. L., adaptation of Frisbee flight principle to delivery of special ordnance, AIAA 72-982, AIAA 2nd Atmospheric Flight Mechanics Conference, Palo Alto, Calif., USA, Sep. 1972; Yasuda, K., Flight- and aerodynamic characteristics of a flying disk, Japanese Soc. Aero. Space Sci., Vol. 47, No. 547, pp. 16 22, 1999 (in Japanese); www.discgolfassoc.com/history.html www.freestyle.org; www.whatisultimate.com/; www.ultimatehandbook.comlWebpages/History/histdisc.html; Frisbee Dynamics: www.disc-wing.com www.lpl.arizona.edut-rlorenz mae.engr.ucdavis.edut-biosport/frisbee/frisbee.html, expressly incorporated herein by reference in their entirety.
Although the modern Frisbee is perhaps the most familiar and popular flying spinning disc, there are a range of variants on the theme. Aerobie produces the “Epic,” which has a conventional-looking upper surface. However, the cylindrical cavity inside the disc is smaller than most discs, and is offset from the center. The offset permits a suitably narrow region for gripping the disc, but the offset displaces the disc center of mass further from the fingers, and in effect lengthens the arm of the thrower, allowing for a faster launch. Aerobie also produces the Superdisk, a disc with a rather flat spoiler rim (made of a comfortable rubber). This disc is allegedly easier to throw than a conventional Frisbee, but does not go as far. If the claim of easy throwing is true, it is presumably a result of the spoiler aerofoil having a pitch moment coefficient that is small over a wider range of angle of attack.
Because of the thickness of the disc required to suppress the pitch moment, the draggy Frisbee does not permit flights of extreme length. A throwing toy that achieves longer distance was developed by Alan Adler, founder of Aerobie, Inc., (formerly Superflight, Inc.) in Palo Alto, Calif., in the 1970s. The flying ring, most commonly encountered in modern times as the Aerobie, is an attempt to circumvent the flying disc's most salient problem, namely the forward center lift and its resultant pitch-up moment. Almost all aerofoil sections have their center of pressure at the quarter chord point, while the center of mass is at the half chord. A ring-wing gets around this problem in part by pure geometry: it can be considered by crude longitudinal section as two separated wings. While the lift on each wing will act forward of the center of each, if the two wings have a sufficiently short chord, this lift offset will be small compared with the overall diameter of the vehicle. For recreational applications, the diameter of the vehicle relates to ergonomics (size of human hand).
A 30 cm diameter provides a large chord, which makes the trailing wing shorter than the leading wing. The trailing wing is also immersed in the downwash from the leading wing. This has the effect of reducing its effective angle of attack and throwing the ring out of balance by reducing the lift on the trailing wing. One approach that was tried initially in the “Skyro” (the first flying ring sold by Aerobie) was to use a rather symmetric aerofoil, but to have it angled such that the wing formed a cone. The trailing wing therefore was mounted at a higher angle of attack to the freestream flow than was the leading wing, and thus when downwash was taken into account the two were at a comparable angle of attack. The two wings thence had the same lift coefficient and the ring flew in a trimmed condition. However, this tuning (a cone angle of only about 1.5 degrees was necessary) was only strictly correct at one flight speed, and thus a perfectly trimmed condition was only found during a portion of a typical flight. Tuning the vehicle over a range of flight conditions instead needed a carefully selected aerofoil section, which had a lift curve slope higher for outwards (trailing) flow than for inwards (leading wing) flow. The higher lift curve slope therefore compensated for the lower angle of attack, such that the resultant lift coefficients were similar. The aerofoil with this characteristic had a rather severe reflex, almost as if it had to spoilers on its trailing edge. This flying ring is rather thin (˜3 mm) compared with a Frisbee, and thus has much lower drag. As a result, the range achievable with a flying ring is much further—the present record is some 1400 ft. This flight was made by throwing along a ridge (so it may have gained from some updraft lift) although it was terminated prematurely, ending about 1.5 m above the ground by striking a bush. This flight lasted only 7 seconds—much shorter than many boomerang flights; although the lift:drag ratio is very good, the actual lift coefficient is small and thus the flight speed must be fairly high.
As with boomerangs and Frisbees, material selection is important. The ring must be adequately weighted to efficiently extract energy from the throw, and to provide sufficient moment of inertia to remain spinning. An additional consideration in this sort of application is compliance, as a metal Aerobie would be rather unpleasant to catch. The flying ring (and its boomerang counterpart) are constructed with a polycarbonate “backbone” which is placed in a mold into which a lower-density rubber is injected. This combination yields the desired density, as well as the desired compliance and “memory” (the ring can be “tuned” slightly by flexing it-were it perfectly elastic, such adjustments would be impossible).
See, Adler, A., The Evolution & Aerodynamics of the Aerobie Flying Ring, note available at www.aerobie.com; Egerton, (The Honourable Wilbraham, M.A., M.P.), An Illustrated Handbook of Indian Arms, 1880; Frohlich, C., Aerodynamic effects on discus flight, Am. J. Phys. 49(12), 1125-1132, 1981; Zdravkovich, M., A. J. Flaherty, M. G. Pahle, and I. A. Skelhorne, Some aerodynamic aspects of coin-like cylinders, J. Fluid Mechanics, 360, 73-84, 1998; www.aerobie.com; www.discwing.com; flyingproducts.com/; www.innovadiscs.com; www.xzylo.com, expressly incorporated herein by reference.
Boomerangs are conventionally divided, by aerodynamicists, at least-into two classes: returning and straight-flying. The former class are largely recreational, while straight-flying boomerangs (the word is derived from “bumarin,” from an aboriginal tribe in New South Wales) were early hunting weapons, perhaps occasionally used in warfare. These straight-flying boomerangs, sometimes called “kylies” or killing-sticks, are every bit as sophisticated as their returning cousins; they develop appreciable lift in flight, without the moments that lead to a curved trajectory which would be undesirable in a hunting weapon. A key feature to note about boomerangs is the basic shape. In essence the boomerang acts as a propeller—the two (or more) arms act as rotor blades as they spin to force air through the disc described by their rotation. For this to happen there must either be a twist in the boomerang or it must be shaped such that the same side of both arms develops lift.
The essence of boomerang flight is its operation as a propeller, combined with gyroscopic precession to yield circular flight. This results in a combination of forward motion and spin, which gives a vertically asymmetric lift distribution (i.e., the roll moment), to produce a horizontal asymmetry (i.e., the pitch moment) by having an eccentric wing. Naturally the forces are greatest at the wingtip that is moving into the flow; at parts of the rotation where the circular motion cancels out the forward motion, the forces are zero. Although there will be some lift produced where the flow is backwards (depending on the twist or camber of the aerofoil), this will typically be rather low compared with the advancing blade. The advancing side clearly gives an asymmetry about the direction of motion, leading to a roll moment which precesses the boomerang around to make its circular path. This asymmetry decreases with advance ratio: a sufficiently fast-spinning wing will essentially feel no effect of Forward motion. The combination of forward flight plus the forward rotational motion of the upper arm gives more lift on the upper arm than the lower. The net lift therefore acts above the axis, leading to a rolling torque. This rolling torque causes the boomerang to precess its spin axis in the horizontal plane. In addition to an upper/lower pressure asymmetry, there is a left-right asymmetry—this produces a pitch torque which causes the boomerang to “lie down” into a horizontal plane. If it is assumed that the component of relative air velocity along the length of the wing does not contribute to lift (i.e., the spanwise flow), then it can be seen that an eccentric wing develops an asymmetry about the orthogonal axis, and thus a pitch moment. Hess presents a boomerang with 8 radial arms (and thus no eccentricity) which nonetheless still “lies down.” In this instance, the blades are close enough together that the pitch moment is produced by wake effects.
The canonical boomerang is angular or crescent-shaped in planform. In most recreational boomerangs, the two wings are of approximately equal length. For the boomerang to be effective, the two wings must both develop lift, as if the boomerang were a propeller, when it spins in one direction, the direction determining the “handedness” of the boomerang. “Right-handed” boomerangs are thrown with the “upper” surface of the boomerang pointing left—the upper surface points towards the thrower's head, and to the center of the circular flight path.
For both wings to develop lift, their aerodynamic surfaces must be shaped accordingly. In the (rather bad) case where the wings are perfectly flat or at least uncambered, the boomerang must have twist, such that both wings encounter a positive angle of attack and thus generate lift in the same direction. If a thicker aerofoil is used with, e.g., a flat base and a curved upper surface, then lift is positive at zero incidence, and this will apply to both wings. A similar result pertains if the airfoil is cambered in the same direction on both wings. The classic type of boomerang is simply angled, or perhaps has a slightly reflexive “Omega” planform, the shape displacing the center of pressure from the center of mass to yield the desired moments during flight. Some other designs are more radially symmetric, forming a three or more pointed “star” shape (the four-armed cross designs come under this category). Several other permutations are possible, many resembling letters of the alphabet (N, T, W, etc.). A final variant is the Aerobie Orbiter, which has an open triangular planform, allowing it to be caught by placing a hand (or foot) in the “hole.” In this example, there is a twist applied to the tip to manipulate the lift distribution along the span.
Some boomerangs have one arm substantially longer than the other. This is particularly the case with MTA (Maximum Time Aloft) boomerangs, and is also characteristic of many Australian aboriginal examples. The long wing gives a large effective moment arm (much like a slingshot or spear-thrower) to permit a higher launch velocity. Also, providing the stability concerns mentioned earlier can be addressed, the moment of inertia of an asymmetric straightish boomerang will be maximized for a given mass; the rotational kinetic energy is what maintains the hover, which is the most important phase for maximizing flight time. A further point is that a long wing gives a higher aspect ratio, which Newman (1985) points out is a key parameter for maximizing the number of turns made by the rotor before motion stops.
Mass distribution is a critical factor for boomerangs, to adjust the center of mass and to change both the mass and the moment of inertia. Large moments of inertia are generally favorable for longer flights (see below), and for this reason masses are often added to the wing tips.
In a classic boomerang flight, the article is thrown with its principal plane inclined outwards by about 20 degrees from vertical. The projection onto the horizontal plane of its flight is approximately circular, with a diameter of typically 30 m; its flight path is initially inclined such that it climbs perhaps 10 or 20 meters into the air. At the end of its circular arc, its forward motion has decayed and the boomerang falls to the ground. The rising and curved path follows simply from the development of lift: most of the lift is projected onto the horizontal plane, causing the article to veer inwards (i.e., to the left, for a right-handed throw). With the initially high forward velocity, the vertical component of lift exceeds the weight of the boomerang, and causes it to accelerate upwards. The clever part of boomerang design derives from the aerodynamic moments. A circular flight path requires that the spin axis be precessed anticlockwise, as seen from above. This is accomplished by a roll moment which is due to the upper wing experiencing a higher airspeed due to the spin; its circumferential velocity adds to the forward velocity of the boomerang, while the lower wing's circumferential speed subtracts from it. This causes an inwards roll; this incremental angular momentum vector points backwards along the direction of flight. Adding this to the spin angular momentum causes the rotation of the latter in the horizontal plane. Any rotor will experience the same sort of effect—a purely linear wing, for example. A common model for this behavior is a cross-shaped boomerang. The radius of the circular flight depends only on three fixed parameters: the lift coefficient, moment of inertia, and span (Hunt, 1999): RF=(4 I)/ρCLπa4), where RF is the radius of the circular flight, a is the span of the boomerang, CL the lift coefficient, and I the moment of inertia about the spin axis. This result relies on CL being invariant, whereas in reality it will depend upon the angle of attack of the throw (although this is not an easy parameter for a thrower to adjust-generally boomerangs are launched at zero angle of attack).
Classical boomerangs have a tendency to “lie down”, the spin plane is initially inclined 20 degrees from vertical, but over the course of the flight it rotates outwards such that the boomerang is more or less horizontal at the end of the flight. This rotation of the angular momentum vector from near-horizontal to vertical requires a pitch-up moment, in essence due to the lift acting forward of the center of mass. Usually this behavior requires the characteristic angled shape of a boomerang. With such a shape, the apex is forward of the center of mass, and the wings are no longer radial to it, but are eccentric. A simple calculation of the square of net velocity (forward plus spin-induced) shows how the center of pressure moves forward as a result.
Note, however, that not all boomerangs can be explained this way. Hess (1968) shows a radially symmetric 8-bladed boomerang (in essence, a throwing star) which also “lies down”. In this case, the forward displacement of the lift must be due to the trailing side of the boomerang operating in the wake or downwash of the leading side. As with the Chakram or Aerobie flying ring, this downwash reduces the effective angle of attack of the trailing side, and thus the lift force from it. The reduced lift on the trailing side therefore displaces the center of pressure forwards and thus precesses the spin axis towards the vertical. A very crude calculation shows the relative roll and pitch moments as being roughly in the ratio of 2 to 1 (Walker 1897): the spin plane is precessed horizontally through something over 180 degrees, while its precession to near vertical involves around 90 degrees. Since the spin rate decreases through the flight, and the horizontal precession may be as much as 360 degrees or more. Note that the horizontal precession (i.e., the roll moment) is a function of the spin rate and flight speed, which determine the airspeeds over the upper and lower wing, while the pitch moment that precesses the angular momentum vector vertical is largely a geometric property.
As is often observed in a conventional boomerang throw, the first (usually upwards) loop is anticlockwise (as seen from above, for a right-handed boomerang), and then the forward flight slows and the boomerang reverses its flight direction, usually to hit the ground soon thereafter. Without the ground, this reversal completes itself and the boomerang begins a clockwise spiral. Like a coil spring, this spiral maintains a constant radius and “pitch” (i.e., vertical interval between successive loops). Increasing the flight path angle (the angle made by the velocity vector with the horizontal) causes a boomerang to reach its maximum altitude more quickly, yet surprisingly it tends to fall down more quickly too. Increasing the angle of attack increases the lift coefficient, and thus (following the simple model above) makes the radius of the flight path smaller, making a tighter loop. One way of increasing the angle of attack fairly early in the flight is to add weights to the inboard part of the boomerang, i.e., to increase the boomerang mass, without increasing the moment of inertia substantially. This retains the same yaw rate (i.e., the spin axis precesses anticlockwise as before), but the velocity vector is rotated anticlockwise more slowly, since the lift now has to accelerate a larger mass. The difference between the two rates yields a rapidly increasing angle of attack. Increasing the roll angle (i.e., launching the boomerang in a more horizontal plane) causes the lift generated by the boomerang to have a stronger vertical component, accelerating the boomerang into the sky. Increasing the spin rate has only a modest effect. This increases the pitch moment (since the advancing and receding wings of the boomerang have a larger speed difference), but at the same time increases the angular momentum. To first order, then, the effects are the same magnitude and cancel out, although for very high spin rates this will not be the case, and the pitch moment will grow faster.
The aerodynamic properties of boomerangs are of course dictated by the aerofoil shape and the planform as constructed. However, the flight characteristics can be modified significantly by small deformations (twisting and bending) done in the field (“tuning”).
Anhedral and dihedral refer to the angle made by the span of a wing with the horizontal: dihedral wings point upwards from root to tip, forming a “V” shape and typically give an aircraft better stability in roll. Anhedral wings form a “A” and tend to have the opposite effect. Anhedral is used on fighter aircraft which need maneuverability rather than stability and on high-winged transport aircraft which already have substantial “pendulum” stability. Applying dihedral (by flexing the wingtips upwards) on a boomerang tends to have the effect of inducing high, hovering flight. This requires the boomerang to “lay over” more quickly—in other words the pitch-up moment is enhanced. This is probably via the airflow hitting the underside of the wingtip at a steeper angle when the tip is pointing forwards. Conversely, applying anhedral yields a lower flight, with later lay-over. Another adjustment is blade twist, to increase the angle of attack of the blades throughout their revolution. This of course increases the lift coefficient and thereby leads to flight in a tighter circle.
A straight-flying boomerang provides an advancing wing which experiences a higher airspeed, and thus a stronger lift than the receding one, and thus a killing stick thrown in a horizontal plane will flip over. Suppressing this tendency requires nulling the roll moment via tuning of the lift distribution, which is accomplished by twisting the blades such that the outboard part of the span produces negative lift. The majority of the lift force developed by a returning boomerang is expended in providing centripetal acceleration to create the circular trajectory. To first order, the ratio of the horizontal component of lift to the vertical component required to balance its weight is the tangent of the boomerang's inclination to the horizontal, or for 70 degrees, around 2.7. Thus, a boomerang that does not need to make a circular flight can afford to generate three times less lift (and correspondingly three times less drag). The blade twist (perhaps combined with a change in section along the span) yields a net positive lift—the inboard positive lift outweighs the outboard negative. However, the roll moment can be made very small, since this is the integral of the lift at each part of the span multiplied by its distance from the center of gravity, such that the outboard negative lift has greater leverage.
See, Azuma, A., G. Beppu, H. Ishikawa, K. Yasuda, Flight dynamics of the boomerang, part I: Fundamental analysis, Journal of Guidance, Control and Dynamics 27, 545-554, July-August 2004; Bahn, P. G., Flight into pre-history, Nature 373, 562, 1987; Bahn, P. G., Return of the Euro-boomerang, Nature 329, 388, 1987. Battipede, M., Boomerang flight mechanics: Unsteady effects on motion characteristics, Journal of Aircraft 36 No. 4, 689-696, 1990; Beppu, G., H. Ishikawa, A. Azuma, K. Yasuda, Flight dynamics of the boomerang, part II: Effects of initial conditions and geometrical configuration, Journal of Guidance, Control and Dynamics 27, 555-562, July-August 2004; Hall, S., Boom in' rangs launches old toy into new orbit, Smithsonian vol. 15, pp. 118-124, 1984; Hess, F., The aerodynamics of boomerangs, Scientific American 219, 124-136, 1968; Hess, F, A returning boomerang from the Iron Age, Antiquity 47, 303-306, 1973; Hunt, H., Bang up a Boomerang, Millennium Mathematics Project, University of Cambridge (pss.maths.org.uldissue7/features/boomhowto/index.html) January 1999; King, A. L., Project boomerang, Am. J. Phys. 43, No. 9, 770-773, 1975. Luebbers, R. A., Ancient boomerangs discovered in South Australia, Nature 253, 39, 1975; Musgrove, P., Many happy returns, New Scientist 61, 186-189, 24 Jan. 1974; Newman, B. G., Boomerangs, Aerospace, 13-18, December 1985; Sharpe, J. W., The boomerang, Philosophical Magazine vol. 10, 60-67, 1905. Thomas, J., Why boomerangs boomerang (and killing-sticks don't), New Scientist, 838-843, 22 September, 1983; Valde-Nowak, P., A. Nadachowski, and M. Wolsan, Upper Paleolithic boomerang made of a mammoth tusk, Nature 329, 436-438, 1987; Walker, G. T., On boomerangs, Phil. Trans. Roy. Soc. London Series A, 190, 23-41, 1897; Ruhe, B., and E. Darnell, Boomerang: How to Make and Catch It, Workman Publishing, New York, 1985; Jones, P., Boomerang: Behind an Australian Icon, Ten Speed Press, 1997, expressly incorporated herein by reference.